# Category: Enumerative combinatorics

Catalan number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the Frenc
Dominance order
In discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partitions of a positive integer n that plays an importan
Exponential formula
In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the
Double counting (proof technique)
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting t
Higher spin alternating sign matrix
In mathematics, a higher spin alternating sign matrix is a generalisation of the alternating sign matrix (ASM), where the columns and rows sum to an integer r (the spin) rather than simply summing to
Stanley–Wilf conjecture
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, states that the growth rate of every proper permutation class is singly exponential. It
Enumerations of specific permutation classes
In the study of permutation patterns, there has been considerable interest in enumerating specific permutation classes, especially those with relatively few basis elements. This area of study has turn
Formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subt
Dixon's identity
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of thr
Solid partition
In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of is a three-dimensional array of non-negative i
Algebraic enumeration
Algebraic enumeration is a subfield of enumeration that deals with finding exact formulas for the number of combinatorial objects of a given type, rather than estimating this number asymptotically. Me
Labelled enumeration theorem
In combinatorial mathematics, the labelled enumeration theorem is the counterpart of the Pólya enumeration theorem for the labelled case, where we have a set of labelled objects given by an exponentia
Cycle index
In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read of
Schuette–Nesbitt formula
In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after and Cecil J. Nesbitt. The probabilistic version of the Schuette–Nesbitt formula
Stirling numbers and exponential generating functions in symbolic combinatorics
The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic c
Combinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number o
MacMahon's master theorem
In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916)
Eulerian number
In combinatorics, the Eulerian number A(n, m) is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). The
List of partition topics
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are * partition of a set or an ordered partition of a set, * parti
Lobb number
In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced p
Poly-Bernoulli number
In mathematics, poly-Bernoulli numbers, denoted as , were defined by M. Kaneko as where Li is the polylogarithm. The are the usual Bernoulli numbers. Moreover, the Generalization of Poly-Bernoulli num
Prüfer sequence
In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length
Bijective proof
In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective funct
Dedekind number
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) counts the number of monotone boolean f
Alternating sign matrix
In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These m
De Bruijn sequence
In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring (i.e., as a co
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and countin
Combinatorial species
In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures, which allows one to not merely count t
Möbius inversion formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 183
Noncrossing partition
In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The number of noncrossi
Rook polynomial
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in
Enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The
Schröder number
In mathematics, the Schröder number also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner of an grid to the northeast corner using
Wilf equivalence
In the study of permutations and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes.Two permutation classes are Wilf equivalent when they have the same numbers of
Aztec diamond
In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n. Here n is a fixed integer, and the square lattice consi
Superpermutation
In combinatorial mathematics, a superpermutation on n symbols is a string that contains each permutation of n symbols as a substring. While trivial superpermutations can simply be made up of every per
Bertrand's ballot theorem
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be stri
Necklace (combinatorics)
In combinatorics, a k-ary necklace of length n is an equivalence class of n-character strings over an alphabet of size k, taking all rotations as equivalent. It represents a structure with n circularl
Vertex enumeration problem
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the ob
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno , although he was not the first to state or prove the f
Motzkin number
In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin nu
Analytic Combinatorics
Analytic Combinatorics is a book on the mathematics of combinatorial enumeration, using generating functions and complex analysis to understand the growth rates of the numbers of combinatorial objects
Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many
Graph enumeration
In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as
Lattice path
In combinatorics, a lattice path L in the d-dimensional integer lattice of length k with steps in the set S, is a sequence of vectors such that each consecutive difference lies in S. A lattice path ma
Inclusion–exclusion principle
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finit
Ordered Bell number
In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing
Pólya enumeration theorem
The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the numb
Eight queens puzzle
The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, co
Schröder–Hipparchus number
In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses i
Plane partition
In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers (with positive integer indices i and j) that is nonincreasing in both indices. This
Dyson conjecture
In mathematics, the Dyson conjecture (Freeman Dyson ) is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it
Necklace polynomial
In combinatorial mathematics, the necklace polynomial, or Moreau's necklace-counting function, introduced by C. Moreau, counts the number of distinct necklaces of n colored beads chosen out of α avail
Double factorial
In mathematics, the double factorial or semifactorial of a number n, denoted by n‼, is the product of all the integers from 1 up to n that have the same parity (odd or even) as n. That is, For even n,
Fence (mathematics)
In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations: or A fence may be finite, or it may be f
Fuss–Catalan number
In combinatorial mathematics and statistics, the Fuss–Catalan numbers are numbers of the form They are named after N. I. Fuss and Eugène Charles Catalan. In some publications this equation is sometime
Alternating permutation
In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set {1, 2, 3, ..., n} is a permutation (arrangement) of those numbers so that each entry is alternately greater
Proofs That Really Count
Proofs That Really Count: the Art of Combinatorial Proof is an undergraduate-level mathematics book on combinatorial proofs of mathematical identies. That is, it concerns equations between two integer
Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also descri