Permutations are represented in gp as `t_VECSMALL`

s and can be input
directly as `Vecsmall([1,3,2,4])`

or obtained from the iterator
`forperm`

:

? forperm(3, p, print(p)) \\ iterate through S_{3}Vecsmall([1, 2, 3]) Vecsmall([1, 3, 2]) Vecsmall([2, 1, 3]) Vecsmall([2, 3, 1]) Vecsmall([3, 1, 2]) Vecsmall([3, 2, 1])

Permutations can be multiplied via `*`

, raised to some power using
`^`

, inverted using `^(-1)`

, conjugated as
`p * q * p^(-1)`

. Their order and signature is available via
`permorder`

and `permsign`

.

binomial coefficient binom{x}{k}. Here k must be an integer, but x can be any PARI object.

? binomial(4,2) %1 = 6 ? n = 4; vector(n+1, k, binomial(n,k-1)) %2 = [1, 4, 6, 4, 1]

The argument k may be omitted if x = n is a
non-negative integer; in this case, return the vector with n+1
components whose k+1-th entry is `binomial`

(n,k)

? binomial(4) %3 = [1, 4, 6, 4, 1]

The library syntax is `GEN `

.**binomial0**(GEN x, GEN k = NULL)

x-th Fibonacci number.

The library syntax is `GEN `

.**fibo**(long x)

If x is a `t_INT`

, return the binary Hamming weight of |x|. Otherwise
x must be of type `t_POL`

, `t_VEC`

, `t_COL`

, `t_VECSMALL`

, or
`t_MAT`

and the function returns the number of non-zero coefficients of
x.

? hammingweight(15) %1 = 4 ? hammingweight(x^100 + 2*x + 1) %2 = 3 ? hammingweight([Mod(1,2), 2, Mod(0,3)]) %3 = 2 ? hammingweight(matid(100)) %4 = 100

The library syntax is `long `

.**hammingweight**(GEN x)

Gives the number of unrestricted partitions of
n, usually called p(n) in the literature; in other words the number of
nonnegative integer solutions to a+2b+3c+.. .= n. n must be of type
integer and n < 10^{15} (with trivial values p(n) = 0 for n < 0 and
p(0) = 1). The algorithm uses the Hardy-Ramanujan-Rademacher formula.
To explicitly enumerate them, see `partitions`

.

The library syntax is `GEN `

.**numbpart**(GEN n)

Generates the k-th permutation (as a row vector of length n) of the
numbers 1 to n. The number k is taken modulo n!, i.e. inverse
function of `permtonum`

. The numbering used is the standard lexicographic
ordering, starting at 0.

The library syntax is `GEN `

.**numtoperm**(long n, GEN k)

Returns the vector of partitions of the integer k as a sum of positive
integers (parts); for k < 0, it returns the empty set `[]`

, and for k
= 0 the trivial partition (no parts). A partition is given by a
`t_VECSMALL`

, where parts are sorted in nondecreasing order:

? partitions(3) %1 = [Vecsmall([3]), Vecsmall([1, 2]), Vecsmall([1, 1, 1])]

correspond to 3, 1+2 and 1+1+1. The number
of (unrestricted) partitions of k is given
by `numbpart`

:

? #partitions(50) %1 = 204226 ? numbpart(50) %2 = 204226

Optional parameters n and a are as follows:

***** n = *nmax* (resp. n = [*nmin*,*nmax*]) restricts
partitions to length less than *nmax* (resp. length between
*nmin* and nmax), where the *length* is the number of nonzero
entries.

***** a = *amax* (resp. a = [*amin*,*amax*]) restricts the parts
to integers less than *amax* (resp. between *amin* and
*amax*).

? partitions(4, 2) \\ parts bounded by 2 %1 = [Vecsmall([2, 2]), Vecsmall([1, 1, 2]), Vecsmall([1, 1, 1, 1])] ? partitions(4,, 2) \\ at most 2 parts %2 = [Vecsmall([4]), Vecsmall([1, 3]), Vecsmall([2, 2])] ? partitions(4,[0,3], 2) \\ at most 2 parts %3 = [Vecsmall([4]), Vecsmall([1, 3]), Vecsmall([2, 2])]

By default, parts are positive and we remove zero entries unless
amin ≤ 0, in which case nmin is ignored and we fix #X = *nmax*:

? partitions(4, [0,3]) \\ parts between 0 and 3 %1 = [Vecsmall([0, 0, 1, 3]), Vecsmall([0, 0, 2, 2]),\ Vecsmall([0, 1, 1, 2]), Vecsmall([1, 1, 1, 1])] ? partitions(1, [0,3], [2,4]) \\ no partition with 2 to 4 non-zero parts %2 = []

The library syntax is `GEN `

.**partitions**(long k, GEN a = NULL, GEN n = NULL)

Given a permutation x on n elements, return its order.

? p = Vecsmall([3,1,4,2,5]); ? p^2 %2 = Vecsmall([4,3,2,1,5]) ? p^4 %3 = Vecsmall([1,2,3,4,5]) ? permorder(p) %4 = 4

The library syntax is `long `

.**permorder**(GEN x)

Given a permutation x on n elements, return its signature.

? p = Vecsmall([3,1,4,2,5]); ? permsign(p) %2 = -1 ? permsign(p^2) %3 = 1

The library syntax is `long `

.**permsign**(GEN x)

Given a permutation x on n elements, gives the number k such that
x = `numtoperm(n,k)`

, i.e. inverse function of `numtoperm`

.
The numbering used is the standard lexicographic ordering, starting at 0.

The library syntax is `GEN `

.**permtonum**(GEN x)

Stirling number of the first kind s(n,k) (*flag* = 1, default) or
of the second kind S(n,k) (*flag* = 2), where n, k are non-negative
integers. The former is (-1)^{n-k} times the
number of permutations of n symbols with exactly k cycles; the latter is
the number of ways of partitioning a set of n elements into k non-empty
subsets. Note that if all s(n,k) are needed, it is much faster to compute
∑_{k} s(n,k) x^k = x(x-1)...(x-n+1).
Similarly, if a large number of S(n,k) are needed for the same k,
one should use
∑_{n} S(n,k) x^n = (x^k)/((1-x)...(1-kx)).
(Should be implemented using a divide and conquer product.) Here are
simple variants for n fixed:

/* list of s(n,k), k = 1..n */ vecstirling(n) = Vec( factorback(vector(n-1,i,1-i*'x)) ) /* list of S(n,k), k = 1..n */ vecstirling2(n) = { my(Q = x^(n-1), t); vector(n, i, t = divrem(Q, x-i); Q=t[1]; simplify(t[2])); } /* Bell numbers, B_{n}= B[n+1] = sum(k = 0, n, S(n,k)), n = 0..N */ vecbell(N)= { my (B = vector(N+1)); B[1] = B[2] = 1; for (n = 2, N, my (C = binomial(n-1)); B[n+1] = sum(k = 1, n, C[k]*B[k]); ); B; }

The library syntax is `GEN `

.
Also available are **stirling**(long n, long k, long flag)`GEN `

(**stirling1**(ulong n, ulong k)*flag* = 1) and `GEN `

(**stirling2**(ulong n, ulong k)*flag* = 2).