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

0,3

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]

Links

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

Example

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

Maple

b:= proc(n, i, p, k) option remember;
      `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))))
    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 *)

Crossrefs

Row sums of A261836.

Cf. A120733.

Sequence in context: A254542 A202602 A097652 * A349503 A225065 A059211

Adjacent sequences: A261835 A261836 A261837 * A261839 A261840 A261841

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2015

Status

approved