7.4: General Combinatorics Problems - Mathematics

As is often the case, practicing one type of problem at a time is helpful to master the techniques necessary to solve that type of problem. In this section, the problem types from the previous three sections are combined.

EXERCISES 4.4
SET I
1) How many strings of six lower case letters from the English alphabet contain
(quad) a) the letter (a ?)
(quad) b) the letters (a) and (b) in consecutive positions with (a) preceding (b), with all the letters distinct?
(quad) c) the letters (a) and (b), where (a) is somewhere to the left of (b) in the string, with all the letters distinct?
2) Seven women and nine men are on the faculty in the mathematics department at a school.
(quad) a) How many ways are there to select a committee of five members of the department if at least one woman must be on the committee?
(quad) b) How many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee?
3) Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with 6 members if it must have the same number of men and women?
4) The English alphabet contains 21 consonants and 5 vowels. How many strings of 6 lower case letters of the English alphabet contain:
(quad) c) at least 1 vowel?
(quad) d) at least 2 vowels?
5) How many ways are there to select 12 countries in the United Nations to serve on a council if 3 are selected from a block of 45,4 are selected from a block of (57,) and the others are selected from the remaining 69 countries?
6) Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with 6 members if it must have more women than men?
7) How many license plates consisting of three letters followed by three digits contain no letter or digit twice?
8) In the women's tennis tournament at Wimbledon, two finalists, A and B, are competing for the title, which will be awarded to the first player to win two sets. In how many different ways can the match be completed?
9) In the men's tennis tournament at Wimbledon, two finalists, (A) and (B), are competing for the title, which will be awarded to the first player to win three sets. In how many different ways can the match be completed?
10) In how many different ways can a panel of 12 jurors and 2 alternate jurors be chosen from a group of 30 prospective jurors?

SET II
11) A class has 20 students, of which 12 are female and 8 are male. In how many ways can a committee of five students be picked from the class if:
(quad) a) No restrictions are placed on the choice of students
(quad) b) No males are included on the committee
(quad) c) The committee must have three female members and 2 male members
12) A school dance committee is to be chosen from a group of students consisting of six freshman, eight sophomores, twelve juniors and ten seniors. If the committee should consist of two freshman, three sophomores, four juniors and five seniors, how many ways can this be done?
13) A theater company consists of 22 actors - 10 men and 12 women. In the next play, the director needs to cast a leading man, leading lady, supporting male role, supporting female role and eight extras (three men and five women). In how many ways can this play be cast?
14) A hockey coach has 20 players of which twelve play forward, six play defense and two are goalies. In how many ways can the coach pick a lineup consisting of three forwards, two defense players and one goalie?
15) In how many ways can ten students be arranged in a row for a class picture if John and Jane want to stand next to each other and Ed and Sally also want to stand next to each other?
16) In how many ways can the students in the previous problem be arranged if Ed and Sally want to stand next to each other, but John and Jane refuse to stand next to each other?
17) In how many ways can four men and four women be seated in a row of eight seats if:
(quad) a) The first seat is occupied by a man
(quad) b) The first and last seats are occupied by women

SET III
18) The social security number of a person is a sequence of nine digits that are not necessarily distinct. How many social security numbers are possible?
19) A variable name in the BASIC programming language is either a letter of the alphabet or a letter followed by a digit. How many distinct variable names are there in the BASIC language?
20) a) How many even numbers are there between 0 and 100?
b) How many even numbers with distinct digits are there between 0 and
(100 ?)
21) There are six characters - three letters of the English alphabet followed by three digits - which appear on the back panel of a particular brand of printer as an identification number. Find the number of possible identification numbers if
(quad) a) characters can be repeated
22) A sequence of characters is called a palindrome if it reads the same forwards and backwards. For example (mathrm{K} 98 mathrm{EE} 89 mathrm{K}) is an eight character palindrome and (mathrm{K} 98 mathrm{E} 89 mathrm{K}) is a seven character palindrome. A MAN A PLAN A CANAL PANAMA is also a palindrome as are WAS IT A RAT I SAW, TACO CAT, TANGY GNAT, and NEVER ODD OR EVEN. Find the number of nine character palindromes that can be formed using the letters of the alphabet such that no letter appears more than twice in each one.
23) Find the number of ways to form a four-letter sequence using the letters (A),
B, (C, D) and (E) if:
(quad) b) repetitions are not permitted
(quad) c) the sequence contains the letter (A) but repetitions are not permitted
(quad) d) the sequence contains the letter (A) and repetitions are permitted
24) There are 10 members A, B, C, D, E, F, G, H, I and J in a fund raising committee. The first task of the committee is to choose a chairperson, a secretary and a treasurer from this group. No individual can hold more than one office. How many ways can these three positions be filled if:
(quad) a) no one has any objection for holding any of these offices
(quad) b) Cneeds to be the chairperson
(quad) d) (quad) A is not willing to serve as chairperson or secretary
(quad) e) either I or J must be the treasurer
(quad) f) (quad) E or (F) or (G) must hold one of the three offices
25) Find the number of ways of picking each of the following from a standard deck of cards.
(quad) a) a king and a queen
(quad) b) a king or a queen
(quad) c) a king and a red card
(quad) d) a king or a red card
26) There are three bridges connecting two towns A and B. Between towns B and
C there are four bridges. A salesperson needs to travel from A to C via B. Find:
(quad) a) the number of possible choices of bridges from A to (C)
(quad) b) the number of choices for a round trip from A to C and back to A
(quad) c) the number of choices for a round trip if no bridge may be crossed twice
27) A sequence of digits in which each digit is 0 or 1 is called binary number. Eight digit binary numbers are often referred to as "bytes."
(quad) a) How many bytes are possible?
(quad) b) How many bytes begin with 10 and end with (01 ?)
(quad) c) How many bytes begin with 10 but do not end with (01 ?)
(quad) d) How many bytes begin with 10 or end with (01 ?)
28) A group of 12 is to be seated in a row of chairs. How many ways can this be done if:
(quad) a) two people, (A) and (B) must be seated next to each other?
(quad) b) two people (A) and (B) must not be seated next to each other?
29) A variable in the FORTRAN language is a sequence that has at most six characters with the first character being a letter of the alphabet and the remaining characters (if any) being either letters or digits. How many distinct variable names are possible?
30) Four station wagons, five sedans and six vans are to be parked in a row of
15 spaces. Find the nubmer of ways to park the vehicles if:
(quad) a) the station wagons are parked at the beginning, then the sedans, then the vans
(quad) b) vehicles of the same type must be parked together

MIT-Harvard-MSR Combinatorics Seminar

The rate of convergence of a random walk on the Cayley graph of the alternating group A_n with respect to a conjugacy class C with support sup(C) le (1-delta)n is Theta() .

This is a far reaching generalization of a famous result of Diaconis and Shahshahani. In particular it solves an open problem of Diaconis.

The result is obtained by new upper bounds on characters of the symmetric groups, and estimations of the probability of a random standard tableau to satisfy certain conditions.

The estimations on characters imply expansion properties of certain Cayley graphs of the symmetric groups, and results on the decomposition of the conjugacy representation of these groups.

Friday, October 7, 4:15 p.m. MIT, room 2-338

On-line Hypergraph Algorithms, Unique Satisfiability of Horn Clauses, and Logic Programming

A hypergraph H, where one vertex of each hyperedge e is designated the head and the remaining vertices of e form the tail, models a set of (pure) Horn clauses. We present a linear on-line algorithm for finding cycles in such a hypergraph when edges are added on-line, which we employ to obtain an optimal (linear time) algorithm for determining whether a set of (general) Horn clauses is uniquely satisfiable. We also present an efficient on-line algorithm for maintaining the distance from a particular vertex r to all the other vertices of the hypergraph when hyperedges are deleted on-line. We discuss several natural applications of the latter algorithm, including an application to well-founded semantics in logic programming.

Wednesday, November 2, 4:15 p.m. MIT, room 2-338

Face numbers of complexes whose Stanley-Reisner ring has bounded depth

A characterization will be given of the possible numbers of faces of simplicial complexes whose Stanley-Reisner ring has depth k or greater. For k = 0 (no condition) this gives the Kruskal-Katona theorem and for maximal depth (the Cohen-Macaulay case) it specializes to Stanley's theorem. Intermediate cases include a characterization of f-vectors of connected complexes. In a more general version the general theorem also involves the Betti numbers of the complex.

No previous knowledge of the area will be assumed - the old results will be reviewed.

Wednesday, November 9, 4:15 p.m. MIT, room 2-338

An overview of Schubert polynomials

Schubert polynomials are a fascinating family of polynomials related to the geometry of flag manifolds. We present a unified theory of Schubert polynomials for the classical groups in the context of combinatorics and algebraic geometry.

Historically, Schubert polynomials (for type A) were defined by Lascoux and Schutzenberger to simplify computations of intersection multiplicities of Schubert varieties and to provide explicit representatives of the Schubert classes defined by Bernstein, Gelfand and Gelfand. Recently, Schubert polynomials have been studied by algebraic combinatorialists because of their connections with the theory of symmetric functions, representation theory and reduced words.

In this talk, we will present some of the background from algebraic geometry on flag manifolds and their cohomology rings. From the geometry there is a natural construction which defines the Schubert polynomials. Once we have a general definition of these polynomials we can give formulas for Schubert polynomials for each of the root systems of type A, B, C, and D. These formulas are defined in terms of Schur Q-functions, the Haiman correspondence of B_n and D_n reduced words, and Stanley symmetric functions. We conclude with some conjectures for expanding products of Schubert polynomials in special cases, these special cases correspond to analogs of the Pieri rules for Schur functions.

Discrete Math Dinner, November 9, 6:00pm, Cambridge Brewing Company

The cost for graduate students and undergraduates will be held down to $$7 (drinks included). The cost for the rest of us should be less than$$20 per person.

Wednesday, November 16, 4:15 p.m. MIT, room 2-338

The path-cycle symmetric function of a digraph

R. Stanley has generalized the chromatic polynomial of a graph to a symmetric function. Independently, F. Chung and R. Graham have defined a digraph polynomial called the cover polynomial, which has close connections with chromatic polynomials and also rook polynomials. In this paper we imitate Stanley's construction to define a symmetric function generalization of the cover polynomial. We obtain generalizations and analogues of several results of Stanley, Chung and Graham, and others. In addition, we prove a combinatorial reciprocity theorem that unexpectedly ties together a number of results scattered in the literature that previously seemed unrelated, and we define a new symmetric function basis that appears to be a natural counterpart of the polynomial basis $_$.

Wednesday, November 30, 4:15 p.m. MIT, room 2-338

An analogue of the Fourier transform for symmetric functions

Motivated by questions in mathematical physics (notably, the work of Kontsevich), Kapranov and I have studied the following problem. Given S_n-modules a(n), associate to a graph G the vector space a(G) given by tensoring together a copy of a(n) for each vertex of G of valence n, and then taking coinvariants under the automorphism group of G. Now sum a(G) over all graphs with given Euler characteristic. We derive a formula for the dimension of this vector space, using the theory of symmetric functions. This formula may be applied to calculate certain combinations of the Euler characteristics of moduli spaces of Riemann surfaces.

Friday, December 2, 4:15 p.m. MIT, room 2-338

Structure constants for Schubert polynomials

Schubert polynomials originated from the study of flag varieties in algebraic geometry. Recently, many of their properties have been elucidated using combinatorics and algebra. A basic open problem is to give a rule for multiplying two Schubert polynomials, that is give a 'Littlewood-Richardson'-type rule for their structure constants.

In this talk, we will establish a formula that is the analog of Pieri's rule for Schubert polynomials. This had previously been conjectured by Bergeron and Billey. While our methods come from algebraic geometry, the proof we present involves only elementary (albeit complicated!) linear algebra.

Wednesday, December 7, 4:15 p.m. MIT, room 2-338

Jay Goldman (Minnesota and M.I.T.)

Knots, tangles, and signed graphs

The signed graphs of tangles or of tunnel links are two-terminal signed networks. The latter contain the two-terminal passive electrical networks. The conductance across the two terminals of such a signed network is defined, generalizing the classical electrical notion. The conductance is a topological invariant of the corresponding tangle or tunnel link. Series, parallel, and star triangle methods generalized from the electrical context yield the first natural interpretation of the graphical Reidemeister moves. The conductance is sensitive to detecting mirror images and linking. Conway's continued fraction of a rational tangle is a conductance and this leads to an elementary proof of Conway's classification of rational tangles. The conductance is related to evaluations of the Jones and Conway polynomials.

Friday, December 9, 4:15 p.m. MIT, room 2-338

On the Littlewood-Richardson and Murnaghan-Nakayama Rules

The Murnaghan-Nakayama Rule is an explicit combinatorial rule for computing a value of an irreducible character of the symmetric group S_n on a given conjugacy class. The similar Littlewood- Richardson Rule describes the coefficients of an expansion of a skew representation of S_n into irreducibles.

We use noncommutative Schur functions to generalize these rules to a large class of representations of S_n, including those related to stable Schubert/Grothendieck polynomials. Alternative versions of the classical L-R and M-N rules will also be given.

Most of the new results are joint with Curtis Greene.

Wednesday, December 14, 4:15 p.m. MIT, room 2-338

On a 2m by 2n chessboard, the maximum number of nonattacking kings that can be placed is mn. The question here is to determine f(m,n), which is the number of such placements. This question was raised by D. E. Knuth in the case m=n.

Using the transfer matrix method, we first express the generating function of f(m,n) as the sum of the entries of x(I-xLambda)^<-1>, where the transfer matrix Lambda is (m+1) 2^m by (m+1) 2^m$. Then we introduce a partial order on the configurations in such a way that the transfer matrix Lambda becomes upper block triangular, and we discuss the spectra of the diagonal blocks. The result is that for each fixed m, we have f(m,n)=(c_mn+d_m)(m+1)^n+O( heta_m^n) (as n goes to infinity), where | heta_m| < m+1. Finally, we discuss also some related work on the problem by other researchers and some open questions. Friday, December 16, 4:15 p.m. MIT, room 2-338 William Jockusch (Michigan) Some mysterious matrix factorizations I will be presenting some mysterious matrix factorizations which arise in the study of Brauer's Centralizer Algebras and which I would like to better understand. I am hoping that someone in the audience will have seen something like them before or will have some ideas about what is going on. Reiterating what was said earlier in a comment: Your answer is almost correct however you made the mistake of picking the positions twice rather than only once. By choosing the four positions to be occupied by letters and then again choosing which three positions were to be occupied by digits and having chosen said positions out of the original seven, you will have run the risk of picking the same position multiple times designating it to be simultaneously used by letter and digit. Instead, the digits' positions should have been selected out of the remaining unused positions giving a total of$inom<3><3>=1$way in which this step could be accomplished, this being merely$1$way it can be left out of the final product. This leaves the final answer as being: Compare this to a slightly larger problem where you have$26$letters to choose from,$10$digits, and$15$special symbols (e.g. [email protected]#$%^ and so on. ) and we want to make a ten-character password consisting of $6$ letters, $3$ digits, and $1$ special symbol with repetition allowed.

We first pick the positions used by letters, then from those remaining positions pick the positions used by digits, letting the final remaining position be occupied by the symbol, and then picking which letters, digits, and symbol they happened to be for a total count of:

Contents

Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person? The answer depends on n.

This has a solution only if n has the form q 2 + q + 1. It is less simple to prove that a solution exists if q is a prime power. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for q congruent to 1 or 2 mod 4, then q is a sum of two square numbers. This last result, the Bruck–Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms.

When such a structure does exist, it is called a finite projective plane thus showing how finite geometry and combinatorics intersect. When q = 2, the projective plane is called the Fano plane.

Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square. One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square. [2]

Combinatorial designs developed along with the general growth of combinatorics from the 18th century, for example with Latin squares in the 18th century and Steiner systems in the 19th century. Designs have also been popular in recreational mathematics, such as Kirkman's schoolgirl problem (1850), and in practical problems, such as the scheduling of round-robin tournaments (solution published 1880s). In the 20th century designs were applied to the design of experiments, notably Latin squares, finite geometry, and association schemes, yielding the field of algebraic statistics.

The classical core of the subject of combinatorial designs is built around balanced incomplete block designs (BIBDs), Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, difference sets, and pairwise balanced designs (PBDs). [3] Other combinatorial designs are related to or have been developed from the study of these fundamental ones.

• A balanced incomplete block design or BIBD (usually called for short a block design) is a collection B of b subsets (called blocks) of a finite set X of v elements, such that any element of X is contained in the same number r of blocks, every block has the same number k of elements, and each pair of distinct elements appear together in the same number λ of blocks. BIBDs are also known as 2-designs and are often denoted as 2-(v,k,λ) designs. As an example, when λ = 1 and b = v, we have a projective plane: X is the point set of the plane and the blocks are the lines.
• A symmetric balanced incomplete block design or SBIBD is a BIBD in which v = b (the number of points equals the number of blocks). They are the single most important and well studied subclass of BIBDs. Projective planes, biplanes and Hadamard 2-designs are all SBIBDs. They are of particular interest since they are the extremal examples of Fisher's inequality (bv).
• A resolvable BIBD is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design. A solution of the famous 15 schoolgirl problem is a resolution of a BIBD with v = 15, k = 3 and λ = 1. [4]
• A Latin rectangle is an r × nmatrix that has the numbers 1, 2, 3, . n as its entries (or any other set of n distinct symbols) with no number occurring more than once in any row or column where rn. An n × n Latin rectangle is called a Latin square. If r < n, then it is possible to append nr rows to an r × n Latin rectangle to form a Latin square, using Hall's marriage theorem. [5]
• A (v, k, λ) difference set is a subsetD of a groupG such that the order of G is v, the size of D is k, and every nonidentity element of G can be expressed as a product d1d2 −1 of elements of D in exactly λ ways (when G is written with a multiplicative operation). [6]
• An Hadamard matrix of order m is an m × m matrix H whose entries are ±1 such that HH ⊤ = mIm, where H ⊤ is the transpose of H and Im is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the order m > 2 then m must be a multiple of 4.
• A pairwise balanced design (or PBD) is a set X together with a family of subsets of X (which need not have the same size and may contain repeats) such that every pair of distinct elements of X is contained in exactly λ (a positive integer) subsets. The set X is allowed to be one of the subsets, and if all the subsets are copies of X, the PBD is called trivial. The size of X is v and the number of subsets in the family (counted with multiplicity) is b.

The Handbook of Combinatorial Designs (Colbourn & Dinitz 2007) has, amongst others, 65 chapters, each devoted to a combinatorial design other than those given above. A partial listing is given below:

Math Major Writing Requirement (Math 300):

If you are a math major, and you would like to complete your major writing requirement through a writing assignment in this class, please let me know in the first week of class and we will discuss it. This writing assignment will not count towards your grade in this class, but will rather just serve as your Major Writing Requirement. If you decide to do this, you must write your paper on a topic in Combinatorics approved by me, and you must keep to a schedule of turning in drafts that is set at the beginning of the semester in order to get credit.

Combinatorial analysis

The branch of mathematics devoted to the solution of problems of choosing and arranging the elements of certain (usually finite) sets in accordance with prescribed rules. Each such rule defines a method of constructing some configuration of elements of the given set, called a combinatorial configuration. One can therefore say that the aim of combinatorial analysis is the study of combinatorial configurations. This study includes questions of the existence of combinatorial configurations, algorithms and their construction, optimization of such algorithms, as well as the solution of problems of enumeration, in particular the determination of the number of configurations of a given class. The simplest examples of combinatorial configurations are permutations, combinations and arrangements.

A set $X$ of $n$ elements is called an $n$- set an $m$- subset of it, $m leq n$, is called a combination of size $m$. The number of combinations of size $m$ from $n$ distinct elements is equal to

$( 1 + t) ^ = sum _ ^ < n >left ( egin n m end ight ) t ^ , left ( egin n 0 end ight ) = 1,$

is usually called the Newton binomial formula. The numbers $C ( n, m)$ are called binomial coefficients. An ordered $m$- subset is called an arrangement of size $m$. The number of arrangements of size $m$ of $n$ distinct elements is equal to

$A ( n, m) = n ( n - 1) dots ( n - m + 1).$

For $m = n$ an arrangement is a permutation of the elements of $X$, the number of such permutations being $P ( n) = n!$.

The rise of the fundamental notions and developments of combinatorial analysis was parallel with the development of other branches of mathematics such as algebra, number theory, probability theory, all closely linked to combinatorial analysis. The formula expressing the number of combinations in terms of the binomial coefficients and the Newton binomial formula for positive integers $n$ was already known to the mathematicians of the Ancient Orient. Magic squares (cf. Magic square) of order three were studied for mystical ends. The birth of combinatorial analysis as a branch of mathematics is associated with the work of B. Pascal and P. (de) Fermat on the theory of games of chance. These works, which formed the foundations of probability theory, contained at the same time the principles for determining the number of combinations of elements of a finite set, and thus established the traditional connection between combinatorial analysis and probability theory.

A large contribution to the systematic development of combinatorial methods was provided by G. Leibniz in his dissertation Ars Combinatoria (the Art of Combinatorics) in which, apparently, the term "combinatorial" appeared for the first time. Of great significance for the establishment of combinatorial analysis was the paper Ars Conjectandi (the Art of Conjecturing) by J. Bernoulli it was devoted to the basic notions of probability theory, and a number of combinatorial notions were of necessity set forth and applications to the calculation of probabilities were given. It can be said that with the appearance of the works of Leibniz and Bernoulli, combinatorial methods started to be an independent branch of mathematics.

A major contribution to the development of combinatorial methods was provided by L. Euler. In his papers on the partitioning and decomposition of positive integers into summands he laid down the beginnings of one of the basic methods of calculating combinatorial configurations, namely the method of generating functions.

The 1950s are associated with an expansion of interest in combinatorial analysis in connection with the rapid development of cybernetics and discrete mathematics and the wide application of computer techniques. In this period an interest in classical combinatorial problems was activated.

Two factors proved to be of influence in the formation of the direction of subsequent investigations. On the one hand, the choice of the objects of investigation on the other, the formation of the aims of the investigation, depending in the final reckoning on the complexity of the objects under study. If the combinatorial configuration under investigation has a complex character, then the aim of the investigation is to elucidate the conditions for its existence and to develop algorithms for its construction.

A large well-developed branch of combinatorial analysis is the theory of block designs (cf. Block design, and also [2], [3], [10]) the main problems of this branch relate to questions of classification, conditions of existence and methods of constructing certain classes of block designs. A special case of block designs are the so-called balanced incomplete block designs or $( b, v, r, k, lambda )$- configurations, which are defined as collections of $b$ $k$- subsets of some $v$- set, called blocks, with the condition that each element appears in $r$ blocks and each pair of elements in $lambda$ blocks. When $b = v$, and hence when $r = k$, a $( b, v, r, k, lambda )$- configuration is called a $( v, k, lambda )$- configuration, or a symmetric balanced incomplete block design. Even for $( v, k, lambda )$- configurations the question of necessary and sufficient conditions for their existence remains unsolved (1988). For the existence of $( v, k, lambda )$- configurations it is necessary that when $v$ is even, $k - lambda$ be a perfect square, while when $v$ is odd, the equation

$z ^ <2>= ( k - lambda ) x ^ <2>+ (- 1) ^ <( v - 1)/2 >lambda y ^ <2>$

must have an integral solution in $x, y, z$, not all zero.

When $v = n ^ <2>+ n + 1$, $k = n + 1$, $lambda = 1$ a $( v, k, lambda )$- configuration represents a projective plane of order $n$ and is a particular case of a finite geometry containing a finite number of points and lines under prescribed incidence conditions. Corresponding to each projective plane of order $n$ there is a unique complete set of $n - 1$ pairwise orthogonal Latin squares of order $n$( cf. Latin square). In order that a projective plane of order $n$ exists, it is necessary that for $n equiv 1, 2$( $mathop < m mod>4$) there exist integers $a, b$ such that

An affirmative solution to the question of the existence of projective planes of order $n$ has been obtained only for $n = p ^ alpha$, where $p$ is a prime number and $alpha$ is a positive integer. Even for $n = 10$ this question remains open (1988). Related to this circle of questions is a result in connection with the falsification of the Euler conjecture on the non-existence of pairs of orthogonal Latin squares of order $n = 4k + 2$, $k = 1, 2 , . . .$( see Classical combinatorial problems).

Another direction in combinatorial analysis relates to selection theorems. At the foundation of a whole series of results along these lines is the P. Hall theorem on the existence of a system of distinct representatives (a transversal) of a family of subsets $( X _ <1>dots X _ )$ of a set $X$, that is, a system of elements $( x _ <1>dots x _ )$ such that $x _ in X _$ and $x _ eq x _$ when $i eq j$. A transversal exists if and only if for any $i _ <1>dots i _$ such that $1 leq i _ <1>< dots < i _ leq n$, $1 leq k leq n$, the following inequality holds:

$| X _ > cup dots cup X _ > | geq k,$

where $| Y |$ is the number of elements in $Y$. A corollary of Hall's theorem is the theorem on the existence of Latin squares, stating that any Latin rectangle of order $k imes n$, $1 leq k leq n - 1$, can be extended to a Latin square of order $n$. Another corollary of Hall's theorem: Any non-negative matrix $A = | a _ | _ ^$ such that

can be represented in the form

$A = alpha _ <1>Pi _ <1>+ dots + alpha _ Pi _ ,$

where $Pi _ <1>dots Pi _$ are permutation matrices of order $n$ and $s leq ( n - 1) ^ <2>+ 1$. Hall's theorem also implies that the minimum number of rows and columns of a non-negative matrix containing all positive elements is equal to the maximum number of elements that pairwise are not in the same row or in the same column. The extremal property of partially ordered sets, which is analogous to this theorem, is established by the theorem stating that the minimum number of non-intersecting chains is the same as the size of the maximal subset consisting of pairwise-incomparable elements. The following theorem also bears an extremal character: If for an $n$- set $X$ one collects all the $C ( n, r)$ combinations of $r$ elements and partitions them into $k$ non-intersecting classes, then, given an integer $m$, there exists an $n _ <0>= n _ <0>( m, r, k)$ such that for $n geq n _ <0>$ there is a subset of $m$ elements $Y subset X$ for which all the $C ( m, r)$ combinations belong to the same class.

The travelling-salesman problem is an extremal problem too it consists in composing the shortest route visiting $n$ towns and returning to the starting point, where the distances between the towns are known. This problem has applications in the study of transportation networks. Combinatorial problems of an extremal character are considered in the theory of flows in networks and in graph theory.

A significant portion of combinatorial analysis consists of enumeration problems. For their solution one either indicates a method of sorting out combinatorial configurations of a given class, or one determines the number of them, or one does both. Typical results of enumeration problems are: The number of permutations of order $n$ with $k$ cycles is equal to $| S ( n, k) |$, where $S ( n, k)$ is the Stirling number of the first kind, defined by the equation

$x ( x - 1) dots ( x - n + 1) = sum _ ^ < n >S ( n, k) x ^$

the number of partitions of a set of $n$ elements into $k$ subsets is equal to the Stirling number of the second kind

$sigma ( n, k) = < frac<1> > sum _ ^ < k >(- 1) ^ left ( egin k j end ight ) ( k - j) ^$

and, the number of arrangements of $m$ distinct objects in $n$ distinct cells with no cell empty is equal to $n! sigma ( m, n)$.

A useful device for the solution of enumeration problems is the permanent of a matrix. The permanent of a matrix $A = | a _ |$( $i = 1 dots n$ $j = 1 dots m$, $n leq m$) the elements of which belong to some ring, is defined by the formula

$mathop < m per>A = sum _ <( j _ <1>dots j _ ) > a _ <1j _ <1>> dots a _ > ,$

where the summation is carried out over all possible arrangements of size $n$ from $m$ distinct elements. The number of transversals of some family of subsets of a finite set is equal to the permanent of the corresponding incidence matrix.

A whole class of problems on the determination of the number of permutations with restricted positions reduces to the calculation of permanents. For convenience, these problems are sometimes formulated as problems on the arrangement of mutually non-attacking pieces on an $n imes n$ chessboard. Connected with the determination of the permanents of certain classes of matrices are variants of the problem of dimers, which arises in the study of the phenomenon of adsorption and consists in the determination of the number of ways of combining the atoms of di-atomic molecules on some surface. Its solution can also be obtained in terms of Pfaffians (cf. Pfaffian), which are certain functions of matrices close to determinants. The problem of the number of Latin rectangles (squares) is also connected with the development of effective methods for calculating permanents of certain $( 0, 1)$- matrices.

For the calculation of permanents one applies the formula:

$mathop < m per>A = sum _ ^ < m >(- 1) ^ left ( egin k m - n end ight ) S _ ,$

$S _ = sum _ <1 leq j _ <1>< dots < j _ leq m > prod _ ^ < n >left ( sum _ ^ < m >a _ - sum _ ^ < k >a _ > ight ) .$

There are a large number of inequalities giving an estimate of the size of the permanent in certain classes of matrices. The determination of the extremal values of the permanent in specific classes of non-negative matrices is of interest. For a $( 0, 1)$- matrix $A$ with given values $r _ <1>dots r _$ of the number of ones in the rows one has the estimate

cf. [12]. The famous van der Waerden conjecture, that the minimum permanent of a doubly-stochastic matrix of order $n$ is equal to $n!/n ^$ was proved, independently, by D.I. Falikman (1979) and G.P. Egorichev (1980), cf. [13].

An important role in the solution of enumeration problems is played by the method of generating functions (cf. Generating function). A generating function

$f ( t) = sum _ ^ infty a _ t ^$

sets up a correspondence between the sequence $( a _ <0>, a _ <1>, . . . )$ and the elements of some ring, and is regarded as a formal power series. According to this definition, generating functions are effectively used for the solution of enumeration problems in parallel with methods of recurrence relations and finite-difference equations. In obtaining asymptotic formulas for the generating functions, analytic functions of a real or complex variable are usually employed. In the latter case, the Cauchy integral is applied in finding expressions for the coefficients.

There are results in the direction of a possible unification of enumeration methods these are connected with the study of so-called incidence algebras and the use of the Möbius function on a partially ordered set (see for example, [10]). In the solution of enumeration problems, an essential role is played by the formalization of the concept of indistinguishability of objects. The use of the notion of equivalence of objects with respect to a certain group of permutations in combination with the application of the method of generating functions forms the basis of the so-called Pólya theory of enumeration (see [10]), the essence of which is as follows. Consider the set $Y ^$ of configurations

$f: X ightarrow Y, | X | = m, | Y | = n.$

On the set $X$, a group $A$ of permutations acts, thus defining an equivalence relation $sim$ under which $f sim f _ <1>$, $f, f _ <1>in Y ^$, if there exists an $alpha in A$ such that $f ( alpha ( x)) = f _ <1>( x)$ for all $x in X$. To each $y in Y$ corresponds a characteristic $[ y] = ( s _ <1>dots s _ )$, where $s _$, $i = 1 dots k$, are elements of an Abelian group. The characteristic of the configuration $f$ is given by the formula

If $a ( s _ <1>dots s _ )$ is the number of elements $y in Y$ with a given value of the characteristic and $b _ ( s _ <1>dots s _ )$ is the number of inequivalent configurations $f in Y ^$,

$F ( y _ <1>dots y _ ) = sum _ <( s _ <1>dots s _ ) > a ( s _ <1>dots s _ ) y _ <1>^ > dots y _ ^ > ,$

$Phi _ ( y _ <1>dots y _ ) = sum _ <( s _ <1>dots s _ ) > b ( s _ <1>dots s _ ) y _ <1>^ > dots y _ ^ > ,$

then Pólya's fundamental theorem states that

$= Z ( F ( y _ <1>dots y _ ), F ( y _ <1>^ <2>dots y _ ^ <2>) dots F ( y _ <1>^ dots y _ ^ )),$

where $Z$ is the cyclic index of the group $A$, defined by the equation

$= sum _ + 2j _ <2>+ dots + mj _ = n > C ( j _ <1>dots j _ A) t _ <1>^ > dots t _ ^ > ,$

and $C ( j _ <1>dots j _ A)$ is the number of elements of the cyclic class $< 1 ^ > dots m ^ > >$( cf. Symmetric group) of $A$. This theorem is based on Burnside's lemma: The number of equivalence classes $N ( A)$ defined on the set $X$ by the permutation group $A$ is given by the formula

where $j _ <1>( alpha )$ is the number of unit cycles of $alpha in A$. Pólya's theory has applications in the solution of enumeration problems in graph theory and in the enumeration of carbon chemical compounds. There is a generalization of Pólya's theory to the case when the equivalence of two configurations is defined by two groups $G$ and $H$ acting on $X$ and $Y$, respectively (see [4] and [10]). In this form it is applied, for example, in the determination of the number of non-isomorphic abstract automata.

If $X = < 1 dots m >$, $Y = < a _ <1>dots a _ >$ and $sigma : X ightarrow Y$, where $a _$ is used as an image $alpha _$ times, then the expression

$[ sigma ] = [ a _ <1>^ > dots a _ ^ > ] , alpha _ <1>+ dots + alpha _ = m,$

is called the first specification of $sigma$. If the numbers $alpha _ <1>dots alpha _$ contain $eta _ <0>$ zeros, $eta _ <1>$ ones, etc., then the expression

$eta _ <1>+ 2 eta _ <2>+ dots + m eta _ = m,$

is called the second specification. Under some specification of the groups $G$ and $H$ defining the equivalence of configurations $sigma : X ightarrow Y$, it is possible to give a method of constructing generating functions for the enumeration of the inequivalent configurations. This method, called the general combinatorial scheme, can be subdivided into four particular cases, according as the groups $G$ and $H$ take values in the identity group $E$ or the symmetric groups $S _$ of corresponding orders. These particular cases are the models for the majority of the known combinatorial schemes (see [9], [10]).

1) The commutative non-symmetric case: $G = S _$, $H = E$. This models combination schemes of distributing identical objects into different cells, etc. The generating function for the enumeration of inequivalent configurations $sigma$ such that

$alpha _ in Lambda _ subseteq mathbf N _ <0>= < 0, 1 , . . . >, Lambda = ( Lambda _ <1>dots Lambda _ ),$

$Phi ( t x _ <1>dots x _ Lambda ) = prod _ ^ < n > sum _ in Lambda _ > ( tx _ ) ^ > .$

2) The non-commutative non-symmetric case: $G = E$, $H = E$. This models allocation schemes of distributing distinct objects into different cells, etc. The generating function for the enumeration of inequivalent configurations $sigma$ such that

$[ sigma ] = [ a _ <1>^ > dots a _ ^ > ] , alpha _ in Lambda _ , Lambda = ( Lambda _ <1>dots Lambda _ ),$

$widetilde Phi ( t x _ <1>dots x _ Lambda ) = prod _ ^ < n > sum _ in Lambda _ > frac > > ! > x _ ^ > .$

3) The commutative symmetric case: $G = S _$, $H = S _$. This models schemes of distributing identical objects into identical cells, the enumeration of the partitions of natural numbers, etc. The enumeration of configurations $sigma$ such that

$[[ sigma ]] = [ [ 0 ^ <eta _ <0>> dots m ^ <eta _ > ] ] , eta _ in Lambda _ , Lambda = ( Lambda _ <1>, Lambda _ <2>, . . . ),$

is based on the use of generating functions of the form:

$Psi ( t x _ <1>, . . . Lambda ) = prod _ ^ infty sum _ <eta _ in Lambda _ > ( x _ t ^ ) ^ <eta _ > .$

4) The non-commutative symmetric case: $G = E$, $H = S _$. This models schemes of partitioning finite sets into blocks, distributing distinct objects into identical cells, etc. The enumeration of configurations $sigma$ such that

$[[ sigma ]] = [ [ 0 ^ <eta _ <0>> dots m ^ <eta _ > ] ] , eta _ in Lambda _ , Lambda = ( Lambda _ <1>, Lambda _ <2>, . . . ),$

is based on the use of generating functions of the form:

$widetilde Psi ( t x _ <1>, . . . Lambda ) = prod _ ^ infty sum _ <eta _ in Lambda _ > left ( frac t ^ > ight ) ^ <eta _ > < frac<1> <eta _ ! > > .$

An important place in combinatorial analysis is taken up by asymptotic methods. They are applied both for the simplification of complex finite expressions for large values of the parameters entering into them, as well as for obtaining approximate formulas in roundabout ways when the exact formulas are unknown. It is sometimes convenient to formulate a combinatorial problem of an enumerative character as a problem of finding the characteristics of the distribution of some random process. Such an interpretation makes it possible to apply the well-developed apparatus of probability theory for finding asymptotics or limit theorems. Classical schemes of random allocations of objects in cells are open to a detailed investigation from these points of view so also are random partitions of sets, the cyclic structure of random permutations, as well as various classes of random graphs, including graphs of mappings (see [8], [9], [11]).

The probabilistic approach is applied in the study of the combinatorial properties of symmetric groups and semi-groups. The limiting distribution of the order of a random element of the symmetric group $S _$ as $n ightarrow infty$ has been investigated, as also have the asymptotics of the probability of the generation of random elements of them. For certain classes of random non-negative matrices, the distributions have been studied of the number of zero rows in a matrix and in permanents, and estimates have been given of the probability of the primitiveness of such matrices. For the proof of the existence of combinatorial configurations without constructing them, one sometimes employs a certain specific probabilistic device. The essence of this device consists in the proof of the existence of the configuration (without constructing it) by means of an estimate of the probability of some event (see [7]).

References

 [1] J. Riordan, "An introduction to combinational analysis" , Wiley (1958) [2] H.J. Ryser, "Combinatorial mathematics" , Carus Math. Monogr. , 14 , Math. Assoc. Amer. (1963) [3] M. Hall, "Combinatorial theory" , Blaisdell (1967) [4] E.F. Beckenbach (ed.) , Applied combinatorial mathematics , Wiley (1964) [5] F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9 [6] F Harary, E. Palmer, "Graphical enumeration" , Acad. Press (1973) [7] P. Erdös, J. Spencer, "Probabilistic methods in combinatorics" , Acad. Press (1974) [8] V.F. Kolchin, V.P. Chistyakov, "Combinatorial problems of probability theory" Itogi Nauk. i Tekhn. Teor. Veroyatnost. Mat. Stat. Teoret. Kibernet. , 11 (1974) pp. 5–45 (In Russian) [9] V.N. Sachkov, , Questions of cybernetics. Proc. Seminar on Combinatorial Mathematics , Moscow (1973) pp. 146–164 (In Russian) [10] V.N. Sachkov, "Combinatorial methods in discrete mathematics" , Moscow (1977) (In Russian) [11] V.N. Sachkov, "Probabilistic methods in combinatorial analysis" , Moscow (1978) (In Russian) [12] H. Minc, "Permanents" , Addison-Wesley (1978) [13] G.P. [G.P. Egorichev] Egorychev, "The solution of van der Waerden's problem for permanents" Adv. in Math. , 42 : 3 (1981) pp. 299–305

The marriage problem is the following. Let there be a set $< g _ <1>dots g _ >$ of $n$ girls and a set $< b _ <1>dots b _ >$ of $m$ boys. Each girl $g _$ likes a subset $B _ subset < b _ <1>dots b _ >$ of boys. When is it possible that each girl marries a boy she likes? The solution is of course given by the P. Hall theorem on distinct representatives, and this theorem is also known as the marriage theorem or the P. Hall marriage theorem. The abbreviation SDR is often used for systems of distinct representatives. Let $G$ be the bipartite graph (cf. Graph, bipartite) consisting of the vertices $< g _ <1>dots g _ , b _ <1>dots b _ >$ and with an edge joining $g _$ and $b _$ if and only if girl $i$ likes boy $j$( and no other edges). Then a solution of the marriage problem provides a matching, and in this context the marriage theorem is also known as the P. Hall matching theorem.

As is the case with all of mathematics, the only way to learn it well is to do as many problems as possible. So, homework problems will be a very important part of the course, and there will be homework assigned every week (other than the week of the midterm). Completion of all homework problems is required, and your grade on a homework assignment will be based on completeness, as well as on the details of the solutions of the problems graded. In particular, I will not necessarily grade every homework problem assigned, but part of your score for an assignment will be for the completion of all problems. Individual homework assignment should be completed by the student alone, although I am always open for questions, either in office hours or by email.

For each homework problem assigned, a complete solution with each step explained should be written up clearly and neatly. Be sure to completely explain your steps and reasoning for calculations as well as for proofs. This is especially important in enumerative problems, as there can be many ways to arrive at the same answer, and what I am interested in is your thought process.

Homework is due at the beginning of class on the due date of the assignment. Late homework will be marked off 20% for every day late. Homework turned in after class on the due date is considered one day late, and the next weekday after that 2 days late, and so on. Everything is easier, of course, if you turn in the homework on time!

 Assignment Problems Due Date 1 5.1 #18, 24, 5.2 #46, 67, 5.3 #14, 19, 5.4 #14, 32 Mon, Jan 30 2 5.5 #14(d,e,f,g), 21, 26, 32, 8.1 #20, 27, 8.2 #10, 25 Mon, Feb 6 3 8.2: Finish the proof of Theorem 2, by proving the formula for Nm * , and #37, 1.1 #16, 18, 22, 1.2 #6, 11, 14 Mon, Feb 13 4 1.3 #9, 14, 15, 1.4 #8, 9, 12, 14, 20 Mon, Feb 20 5 2.1 #8, 12(a,c), 13, 14, 2.2 #4(b,h), 6, 16 Mon, Feb 27 6 2.3 #1(d,f), 2.4 #1, 2, 8(a,b), 11(a,b,c) Wed, Mar 14 7 6.1 #14, 22, 6.2 #18, 31, 6.3 #2, 15, 18, 20 Fri, Apr 6 8 6.3 #21, 6.4 #2, 12, 20, 6.5 #1(c,d,e), #2(c,d,e) Mon, Apr 16 9 7.3 #3(b,c,d), 5, 7, 7.4 #6, 9(b,c,d), 7.5 #1(b,c), #2(for #1b,c) Mon, Apr 23

Contents

Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, [3] selection of investments and portfolios, [4] selection of assets for asset-backed securitization, [5] and generating keys for the Merkle–Hellman [6] and other knapsack cryptosystems.

One early application of knapsack algorithms was in the construction and scoring of tests in which the test-takers have a choice as to which questions they answer. For small examples, it is a fairly simple process to provide the test-takers with such a choice. For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve a maximum possible score of 100 points. However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices. Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points. The students are asked to answer all of the questions to the best of their abilities. Of the possible subsets of problems whose total point values add up to 100, a knapsack algorithm would determine which subset gives each student the highest possible score. [7]

A 1999 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem. [8]

The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number x i > of copies of each kind of item to a maximum non-negative integer value c :

The unbounded knapsack problem (UKP) places no upper bound on the number of copies of each kind of item and can be formulated as above except for that the only restriction on x i > is that it is a non-negative integer.

One example of the unbounded knapsack problem is given using the figure shown at the beginning of this article and the text "if any number of each box is available" in the caption of that figure.

The knapsack problem is interesting from the perspective of computer science for many reasons:

• The decision problem form of the knapsack problem (Can a value of at least V be achieved without exceeding the weight W?) is NP-complete, thus there is no known algorithm both correct and fast (polynomial-time) in all cases.
• While the decision problem is NP-complete, the optimization problem is not, its resolution is at least as difficult as the decision problem, and there is no known polynomial algorithm which can tell, given a solution, whether it is optimal (which would mean that there is no solution with a larger V, thus solving the NP-complete decision problem).
• There is a pseudo-polynomial time algorithm using dynamic programming.
• There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine, described below.
• Many cases that arise in practice, and "random instances" from some distributions, can nonetheless be solved exactly.

There is a link between the "decision" and "optimization" problems in that if there exists a polynomial algorithm that solves the "decision" problem, then one can find the maximum value for the optimization problem in polynomial time by applying this algorithm iteratively while increasing the value of k. On the other hand, if an algorithm finds the optimal value of the optimization problem in polynomial time, then the decision problem can be solved in polynomial time by comparing the value of the solution output by this algorithm with the value of k. Thus, both versions of the problem are of similar difficulty.

One theme in research literature is to identify what the "hard" instances of the knapsack problem look like, [9] [10] or viewed another way, to identify what properties of instances in practice might make them more amenable than their worst-case NP-complete behaviour suggests. [11] The goal in finding these "hard" instances is for their use in public key cryptography systems, such as the Merkle-Hellman knapsack cryptosystem.

Furthermore, notable is the fact that the hardness of the knapsack problem depends on the form of the input. If the weights and profits are given as integers, it is weakly NP-complete, while it is strongly NP-complete if the weights and profits are given as rational numbers. [12] However, in the case of rational weights and profits it still admits a fully polynomial-time approximation scheme.

Several algorithms are available to solve knapsack problems, based on the dynamic programming approach, [13] the branch and bound approach [14] or hybridizations of both approaches. [11] [15] [16] [17]

The unbounded knapsack problem (UKP) places no restriction on the number of copies of each kind of item. Besides, here we assume that x i > 0 >0>

History

Certain types of combinatorial problems have attracted the attention of mathematicians since early times. Magic squares, for example, which are square arrays of numbers with the property that the rows, columns, and diagonals add up to the same number, occur in the I Ching, a Chinese book dating back to the 12th century bc . The binomial coefficients, or integer coefficients in the expansion of (a + b) n , were known to the 12th-century Indian mathematician Bhāskara, who in his Līlāvatī (“The Graceful”), dedicated to a beautiful woman, gave the rules for calculating them together with illustrative examples. “Pascal’s triangle,” a triangular array of binomial coefficients, had been taught by the 13th-century Persian philosopher Naṣīr ad-Dīn aḷ-Ṭūsī.

In the West, combinatorics may be considered to begin in the 17th century with Blaise Pascal and Pierre de Fermat, both of France, who discovered many classical combinatorial results in connection with the development of the theory of probability. The term combinatorial was first used in the modern mathematical sense by the German philosopher and mathematician Gottfried Wilhelm Leibniz in his Dissertatio de Arte Combinatoria (“Dissertation Concerning the Combinational Arts”). He foresaw the applications of this new discipline to the whole range of the sciences. The Swiss mathematician Leonhard Euler was finally responsible for the development of a school of authentic combinatorial mathematics beginning in the 18th century. He became the father of graph theory when he settled the Königsberg bridge problem, and his famous conjecture on Latin squares was not resolved until 1959.

In England, Arthur Cayley, near the end of the 19th century, made important contributions to enumerative graph theory, and James Joseph Sylvester discovered many combinatorial results. The British mathematician George Boole at about the same time used combinatorial methods in connection with the development of symbolic logic, and the combinatorial ideas and methods of Henri Poincaré, which developed in the early part of the 20th century in connection with the problem of n bodies, have led to the discipline of topology, which occupies the centre of the stage of mathematics. Many combinatorial problems were posed during the 19th century as purely recreational problems and are identified by such names as “the problem of eight queens” and “the Kirkman school girl problem.” On the other hand, the study of triple systems begun by Thomas P. Kirkman in 1847 and pursued by Jakob Steiner, a Swiss-born German mathematician, in the 1850s was the beginning of the theory of design. Among the earliest books devoted exclusively to combinatorics are the German mathematician Eugen Netto’s Lehrbuch der Combinatorik (1901 “Textbook of Combinatorics”) and the British mathematician Percy Alexander MacMahon’s Combinatory Analysis (1915–16), which provide a view of combinatorial theory as it existed before 1920.

Contents

Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 2 6 − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers. [7] [8] [9] Earlier, in the Ostomachion, Archimedes (3rd century BCE) may have considered the number of configurations of a tiling puzzle, [10] while combinatorial interests possibly were present in lost works by Apollonius. [11] [12]

In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra (c. 850) provided formulae for the number of permutations and combinations, [13] [14] and these formulas may have been familiar to Indian mathematicians as early as the 6th century CE. [15] The philosopher and astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321. [16] The arithmetical triangle—a graphical diagram showing relationships among the binomial coefficients—was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. [17] [18]

During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics. Graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem.

In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject. [19] In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field.

Enumerative combinatorics Edit

Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.

Analytic combinatorics Edit

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.

Partition theory Edit

Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory and has connections with statistical mechanics.

Graph theory Edit

Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on n vertices with k edges) to existing structures (e.g., Hamiltonian cycles) to algebraic representations (e.g., given a graph G and two numbers x and y, does the Tutte polynomial TG(x,y) have a combinatorial interpretation?). Although there are very strong connections between graph theory and combinatorics, they are sometimes thought of as separate subjects. [20] While combinatorial methods apply to many graph theory problems, the two disciplines are generally used to seek solutions to different types of problems.

Design theory Edit

Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics.

Finite geometry Edit

Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory. It should not be confused with discrete geometry (combinatorial geometry).

Order theory Edit

Order theory is the study of partially ordered sets, both finite and infinite. Various examples of partial orders appear in algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and Boolean algebras.

Matroid theory Edit

Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory. It is now an independent field of study with a number of connections with other parts of combinatorics.

Extremal combinatorics Edit

Extremal combinatorics studies extremal questions on set systems. The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on 2n vertices is a complete bipartite graph Kn,n. Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate.

Ramsey theory is another part of extremal combinatorics. It states that any sufficiently large configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle.

Probabilistic combinatorics Edit

In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find), simply by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as the probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time.

Often associated with Paul Erdős, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. However, with the growth of applications to analyze algorithms in computer science, as well as classical probability, additive number theory, and probabilistic number theory, the area recently grew to become an independent field of combinatorics.

Algebraic combinatorics Edit

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, and can be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant.

Combinatorics on words Edit

Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory, and linguistics. While many applications are new, the classical Chomsky–Schützenberger hierarchy of classes of formal grammars is perhaps the best-known result in the field.

Geometric combinatorics Edit

Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. Combinatorial geometry is an old fashioned name for discrete geometry.

Topological combinatorics Edit

Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.

Arithmetic combinatorics Edit

Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory, and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems.

Infinitary combinatorics Edit

Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics.

Gian-Carlo Rota used the name continuous combinatorics [21] to describe geometric probability, since there are many analogies between counting and measure.

Combinatorial optimization Edit

Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory.

Coding theory Edit

Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of information theory.

Discrete and computational geometry Edit

Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.

Combinatorics and dynamical systems Edit

Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system.

Combinatorics and physics Edit

There are increasing interactions between combinatorics and physics, particularly statistical physics. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic and Tutte polynomials on the other hand.