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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. 3
1, 1, 2, 20, 48, 264, 4296, 14528, 89472, 593248, 19115360, 75604544, 599169408, 4141674240, 40147321344, 2159264715776, 10240251475456, 92926573965184, 746025520714112, 7285397378650112, 82900557619046912, 7796186873306241024, 41825012467664893440 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 25 2017, translated from Maple *)

CROSSREFS

Row sums of A261836.

Cf. A120733.

Sequence in context: A254542 A202602 A097652 * A225065 A059211 A139271

Adjacent sequences:  A261835 A261836 A261837 * A261839 A261840 A261841

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 02 2015

STATUS

approved

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)