Combinatorial Functions

n!	factorial
n!!	double factorial
Binomial[n,m]	binomial coefficient
Multinomial[n₁,n₂,...]	multinomial coefficient
CatalanNumber[n]	Catalan number
Hyperfactorial[n]	hyperfactorial
BarnesG[n]	Barnes G-function
Subfactorial[n]	number of derangements of objects
Fibonacci[n]	Fibonacci number
Fibonacci[n,x]	Fibonacci polynomial
LucasL[n]	Lucas number
LucasL[n,x]	Lucas polynomial
HarmonicNumber[n]	harmonic number
HarmonicNumber[n,r]	harmonic number of order
BernoulliB[n]	Bernoulli number
BernoulliB[n,x]	Bernoulli polynomial
NorlundB[n,a]	Nörlund polynomial
NorlundB[n,a,x]	generalized Bernoulli polynomial
EulerE[n]	Euler number
EulerE[n,x]	Euler polynomial
StirlingS1[n,m]	Stirling number of the first kind
StirlingS2[n,m]	Stirling number of the second kind
BellB[n]	Bell number
BellB[n,x]	Bell polynomial
PartitionsP[n]	the number of unrestricted partitions of the integer
IntegerPartitions[n]	partitions of an integer
PartitionsQ[n]	the number of partitions of into distinct parts
Signature[{i₁,i₂,...}]	the signature of a permutation

Combinatorial functions.

The factorial function

gives the number of ways of ordering

objects. For non-integer

, the numerical value of

is obtained from the gamma function, discussed in "Special Functions".

The binomial coefficient Binomial

can be written as

. It gives the number of ways of choosing

objects from a collection of

objects, without regard to order. The Catalan numbers, which appear in various tree enumeration problems, are given in terms of binomial coefficients as

The subfactorial Subfactorial[n] gives the number of permutations of

objects that leave no object fixed. Such a permutation is called a derangement. The subfactorial is given by

The multinomial coefficient Multinomial

, denoted

, gives the number of ways of partitioning

distinct objects into

sets of sizes

(with

Mathematica gives the exact integer result for the factorial of an integer.

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For non-integers, Mathematica evaluates factorials using the gamma function.

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Mathematica can give symbolic results for some binomial coefficients.

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This gives the number of ways of partitioning

objects into sets containing 6 and 5 objects.

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The result is the same as

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The Fibonacci numbers Fibonacci[n] satisfy the recurrence relation

with

. They appear in a wide range of discrete mathematical problems. For large

approaches the golden ratio. The Lucas numbers LucasL[n] satisfy the same recurrence relation as the Fibonacci numbers do, but with initial conditions

and

The Fibonacci polynomials Fibonacci

appear as the coefficients of

in the expansion of

The harmonic numbers HarmonicNumber[n] are given by

; the harmonic numbers of order

HarmonicNumber

are given by

. Harmonic numbers appear in many combinatorial estimation problems, often playing the role of discrete analogs of logarithms.

The Bernoulli polynomials BernoulliB

satisfy the generating function relation

. The Bernoulli numbers BernoulliB[n] are given by

. The

appear as the coefficients of the terms in the Euler-Maclaurin summation formula for approximating integrals. The Bernoulli numbers are related to the Genocchi numbers by

Numerical values for Bernoulli numbers are needed in many numerical algorithms. You can always get these numerical values by first finding exact rational results using BernoulliB[n], and then applying N.

The Euler polynomials EulerE

have generating function

, and the Euler numbers EulerE[n] are given by

The Nörlund polynomials NorlundB

satisfy the generating function relation

. The Nörlund polynomials give the Bernoulli numbers when

. For other positive integer values of

, the Nörlund polynomials give higher-order Bernoulli numbers. The generalized Bernoulli polynomials NorlundB[n, a, x] satisfy the generating function relation

This gives the second Bernoulli polynomial

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You can also get Bernoulli polynomials by explicitly computing the power series for the generating function.

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BernoulliB[n] gives exact rational-number results for Bernoulli numbers.

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Stirling numbers show up in many combinatorial enumeration problems. For Stirling numbers of the first kind StirlingS1

gives the number of permutations of

elements which contain exactly

cycles. These Stirling numbers satisfy the generating function relation

. Note that some definitions of the

differ by a factor

from what is used in Mathematica.

Stirling numbers of the second kind StirlingS2

, sometimes denoted

, give the number of ways of partitioning a set of

elements into

non-empty subsets. They satisfy the relation

The Bell numbers BellB[n] give the total number of ways that a set of

elements can be partitioned into non-empty subsets. The Bell polynomials BellB

satisfy the generating function relation

The partition function PartitionsP[n] gives the number of ways of writing the integer

as a sum of positive integers, without regard to order. PartitionsQ[n] gives the number of ways of writing

as a sum of positive integers, with the constraint that all the integers in each sum are distinct.

IntegerPartitions[n] gives a list of the partitions of

, with length

This gives a table of Stirling numbers of the first kind.

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The Stirling numbers appear as coefficients in this product.

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Here are the partitions of 4.

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The number of partitions is given by PartitionsP

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This gives the number of partitions of 100, with and without the constraint that the terms should be distinct.

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The partition function

increases asymptotically like

. Note that you cannot simply use Plot to generate a plot of a function like PartitionsP because the function can only be evaluated with integer arguments.

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Most of the functions here allow you to count various kinds of combinatorial objects. Functions like IntegerPartitions and Permutations allow you instead to generate lists of various combinations of elements.

The signature function Signature

gives the signature of a permutation. It is equal to

for even permutations (composed of an even number of transpositions), and to

for odd permutations. The signature function can be thought of as a totally antisymmetric tensor, Levi-Civita symbol or epsilon symbol.

ClebschGordan[{j₁,m₁},{j₂,m₂},{j,m}]	Clebsch-Gordan coefficient
ThreeJSymbol[{j₁,m₁},{j₂,m₂},{j₃,m₃}]	Wigner 3-j symbol
SixJSymbol[{j₁,j₂,j₃},{j₄,j₅,j₆}]	Racah 6-j symbol

Rotational coupling coefficients.

Clebsch-Gordan coefficients and

-j symbols arise in the study of angular momenta in quantum mechanics, and in other applications of the rotation group. The Clebsch-Gordan coefficients ClebschGordan

give the coefficients in the expansion of the quantum mechanical angular momentum state

in terms of products of states

The 3-j symbols or Wigner coefficients ThreeJSymbol

are a more symmetrical form of Clebsch-Gordan coefficients. In Mathematica, the Clebsch-Gordan coefficients are given in terms of 3-j symbols by

The 6-j symbols SixJSymbol

give the couplings of three quantum mechanical angular momentum states. The Racah coefficients are related by a phase to the 6-j symbols.

You can give symbolic parameters in 3-j symbols.

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