A261838 Number of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet (k=1,2,3,...) whose letters appear in alphabetical order and all k letters occur at least once in the composition.

1, 1, 2, 20, 48, 264, 4296, 14528, 89472, 593248, 19115360, 75604544, 599169408, 4141674240, 40147321344, 2159264715776, 10240251475456, 92926573965184, 746025520714112, 7285397378650112, 82900557619046912, 7796186873306241024, 41825012467664893440

Offset: 0

Author

Alois P. Heinz, Sep 02 2015

Keywords

• nonn

Comments

Also number of matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n and the column sums are distinct.

a(2) = 2:

[1] [2]

[1]

Examples

			a(0) = 1: the empty composition.
a(1) = 1: 1a.
a(2) = 2: 2aa (for k=1), 2ab (for k=2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Row sums of A261836.

Cf. A120733.

Programs

Maple

b:= proc(n, i, p, k) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> add(add(b(n$2, 0, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);

Mathematica

b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; a[n_] := Sum[b[n, n, 0, k-i]*(-1)^i*Binomial[k, i], {k, 0, n}, {i, 0, k}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 25 2017, translated from Maple *)