Combinatorics
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## Combinatorics

Catalan

      Catalan (n)

  Get `n`th Catalan number.

  See Planetmath for more information.

Combinations

      Combinations (k,n)

  Get all combinations of k numbers from 1 to n as a vector of vectors. (See also NextCombination)

  See Wikipedia for more information.

DoubleFactorial

      DoubleFactorial (n)

  Double factorial: `n(n-2)(n-4)...`

  See Planetmath for more information.

Factorial

      Factorial (n)

  Factorial: `n(n-1)(n-2)...`

  See Planetmath for more information.

FallingFactorial

      FallingFactorial (n,k)

  Falling factorial: `(n)_k = n(n-1)...(n-(k-1))`

  See Planetmath for more information.

Fibonacci

      Fibonacci (x)

  Aliases: `fib`

  Calculate `n`th Fibonacci number. That is the number defined recursively by `Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)` and `Fibonacci(1) = Fibonacci(2) = 1`.

  See Wikipedia or Planetmath or Mathworld for more information.

FrobeniusNumber

      FrobeniusNumber (v,arg...)

  Calculate the Frobenius number. That is calculate largest number that cannot be given as a non-negative integer linear combination of a given vector of non-negative integers. The vector can be given as separate numbers or a single vector. All the numbers given should have GCD of 1.

  See Wikipedia or Mathworld for more information.

GaloisMatrix

      GaloisMatrix (combining_rule)

  Galois matrix given a linear combining rule (a_1*x_1+...+a_n*x_n=x_(n+1)).

GreedyAlgorithm

      GreedyAlgorithm (n,v)

  Find the vector `c` of non-negative integers such that taking the dot product with `v` is equal to n. If not possible returns `null`. `v` should be given sorted in increasing order and should consist of non-negative integers.

  See Wikipedia or Mathworld for more information.

HarmonicNumber

      HarmonicNumber (n,r)

  Aliases: `HarmonicH`

  Harmonic Number, the `n`th harmonic number of order `r`. That is, it is the sum of `1/k^r` for `k` from 1 to n. Equivalent to `sum k = 1 to n do 1/k^r`.

  See Wikipedia for more information.

Hofstadter

      Hofstadter (n)

  Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).

  See Wikipedia for more information. The sequence is A005185 in OEIS.

LinearRecursiveSequence

      LinearRecursiveSequence (seed_values,combining_rule,n)

  Compute linear recursive sequence using Galois stepping.

Multinomial

      Multinomial (v,arg...)

  Calculate multinomial coefficients. Takes a vector of `k` non-negative integers and computes the multinomial coefficient. This corresponds to the coefficient in the homogeneous polynomial in `k` variables with the corresponding powers.

  The formula for `Multinomial(a,b,c)` can be written as:

      (a+b+c)! / (a!b!c!)

  In other words, if we would have only two elements, then `Multinomial(a,b)` is the same thing as `Binomial(a+b,a)` or `Binomial(a+b,b)`.

  See Wikipedia, Planetmath, or Mathworld for more information.

NextCombination

      NextCombination (v,n)

  Get combination that would come after v in call to combinations, first combination should be `[1:k]`. This function is useful if you have many combinations to go through and you don't want to waste memory to store them all.

  For example with Combinations you would normally write a loop like:

      for n in Combinations (4,6) do (
        SomeFunction (n)
      );

  But with NextCombination you would write something like:

      n:=[1:4];
      do (
        SomeFunction (n)
      ) while not IsNull(n:=NextCombination(n,6));

  See also Combinations.

  See Wikipedia for more information.

Pascal

      Pascal (i)

  Get the Pascal's triangle as a matrix. This will return an `i`+1 by `i`+1 lower diagonal matrix that is the Pascal's triangle after `i` iterations.

  See Planetmath for more information.

Permutations

      Permutations (k,n)

  Get all permutations of `k` numbers from 1 to `n` as a vector of vectors.

  See Mathworld or Wikipedia for more information.

RisingFactorial

      RisingFactorial (n,k)

  Aliases: `Pochhammer`

  (Pochhammer) Rising factorial: (n)_k = n(n+1)...(n+(k-1)).

  See Planetmath for more information.

StirlingNumberFirst

      StirlingNumberFirst (n,m)

  Aliases: `StirlingS1`

  Stirling number of the first kind.

  See Planetmath or Mathworld for more information.

StirlingNumberSecond

      StirlingNumberSecond (n,m)

  Aliases: `StirlingS2`

  Stirling number of the second kind.

  See Planetmath or Mathworld for more information.

Subfactorial

      Subfactorial (n)

  Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.

Triangular

      Triangular (nth)

  Calculate the `n`th triangular number.

  See Planetmath for more information.

nCr

      nCr (n,r)

  Aliases: `Binomial`

  Calculate combinations, that is, the binomial coefficient. `n` can be any real number.

  See Planetmath for more information.

nPr

      nPr (n,r)

  Calculate the number of permutations of size `r` of numbers from 1 to `n`.

  See Mathworld or Wikipedia for more information.

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