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[n, m] 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[n1, n2, ...], 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[n, x] appear as the coefficients of 
in the expansion of 
.
The harmonic numbers HarmonicNumber[n] are given by 
; the harmonic numbers of order 
HarmonicNumber[n, r] are given by 
. Harmonic numbers appear in many combinatorial estimation problems, often playing the role of discrete analogs of logarithms.
The Bernoulli polynomials BernoulliB[n, x] 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[n, x] have generating function 
, and the Euler numbers EulerE[n] are given by 
.
The Nörlund polynomials NorlundB[n, a] 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[n, m], 
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[n, m], 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[n, x] 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 ![TemplateBox[{PartitionsP, paclet:ref/PartitionsP}, RefLink, BaseStyle -> InlineFormula][n]](Files/CombinatorialFunctions.en/Image_84.png "TemplateBox[{PartitionsP, paclet:ref/PartitionsP}, RefLink, BaseStyle -> InlineFormula][n]")
.
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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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[{i1, i2, ...}] 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[{j1, m1}, {j2, m2}, {j, m}] 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[{j1, m1}, {j2, m2}, {j3, m3}] 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[{j1, j2, j3}, {j4, j5, j6}] 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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