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# Combinatorial Functions

 n! factorial n (n-1) (n-2)×...×1 n!! double factorial n (n-2) (n-4)×... Binomial[n,m] binomial coefficient Multinomial[n1,n2,...] multinomial coefficient (n1+n2+...)!/ (n1!n2!...) CatalanNumber[n] Catalan number cn Hyperfactorial[n] hyperfactorial 11 22... nn BarnesG[n] Barnes G-function 1! 2!... (n-2)! Subfactorial[n] number of derangements of n objects Fibonacci[n] Fibonacci number Fn Fibonacci[n,x] Fibonacci polynomial Fn (x) LucasL[n] Lucas number Ln LucasL[n,x] Lucas polynomial Ln (x) HarmonicNumber[n] harmonic number Hn HarmonicNumber[n,r] harmonic number of order r BernoulliB[n] Bernoulli number Bn BernoulliB[n,x] Bernoulli polynomial Bn (x) NorlundB[n,a] Nörlund polynomial NorlundB[n,a,x] generalized Bernoulli polynomial EulerE[n] Euler number En EulerE[n,x] Euler polynomial En (x) StirlingS1[n,m] Stirling number of the first kind StirlingS2[n,m] Stirling number of the second kind BellB[n] Bell number Bn BellB[n,x] Bell polynomial Bn (x) PartitionsP[n] the number p (n) of unrestricted partitions of the integer n IntegerPartitions[n] partitions of an integer PartitionsQ[n] the number q (n) of partitions of n into distinct parts Signature[{i1,i2,...}] the signature of a permutation

Combinatorial functions.

The factorial function n! gives the number of ways of ordering n objects. For non-integer n, the numerical value of n! 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 m objects from a collection of n 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 n objects that leave no object fixed. Such a permutation is called a derangement. The subfactorial is given by n! (-1)k/k!.
The multinomial coefficient Multinomial[n1, n2, ...], denoted (N;n1, n2, ..., nm)=N!/ (n1!n2!... nm!), gives the number of ways of partitioning N distinct objects into m sets of sizes ni (with N=ni).
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 6+5=11 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 Fn=Fn-1+Fn-2 with F1=F2=1. They appear in a wide range of discrete mathematical problems. For large n, Fn/Fn-1 approaches the golden ratio. The Lucas numbers LucasL[n] satisfy the same recurrence relation as the Fibonacci numbers do, but with initial conditions L1=1 and L2=3.
The Fibonacci polynomials Fibonacci[n, x] appear as the coefficients of tn in the expansion of t/ (1-xt-t2)=Fn (x)tn .
The harmonic numbers are given by Hn=1/i; the harmonic numbers of order r 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 tex t/ (et-1)=Bn (x)tn/n! . The Bernoulli numbers BernoulliB[n] are given by Bn=Bn (0). The Bn 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 Gn=2 (1-2n)Bn.
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 2ex t/ (et+1)=En (x)tn/n! , 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 a=1. For other positive integer values of a, 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 B2 (x).
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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 n elements which contain exactly m cycles. These Stirling numbers satisfy the generating function relation . Note that some definitions of the differ by a factor (-1)n-m 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 n elements into m non-empty subsets. They satisfy the relation .
The Bell numbers BellB[n] give the total number of ways that a set of n 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 n as a sum of positive integers, without regard to order. PartitionsQ[n] gives the number of ways of writing n as a sum of positive integers, with the constraint that all the integers in each sum are distinct.
gives a list of the partitions of n, with length PartitionsP[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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The number of partitions is given by PartitionsP[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 p (n) 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 +1 for even permutations (composed of an even number of transpositions), and to -1 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 n-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 j, m in terms of products of states j1, m1 j2, m2.
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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