Combinatorics

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 S3
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(x, {k})

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)`.

fibonacci(x)

x-th Fibonacci number.

The library syntax is `GEN fibo(long x)`.

hammingweight(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)`.

numbpart(n)

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 < 1015 (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)`.

numtoperm(n, k)

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)`.

partitions(k, {a = k}, {n = 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)`.

permorder(x)

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)`.

permsign(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)`.

permtonum(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(n, k, {flag = 1})

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, Bn = 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 stirling(long n, long k, long flag)`. Also available are `GEN stirling1(ulong n, ulong k)` (flag = 1) and `GEN stirling2(ulong n, ulong k)` (flag = 2).