A332747 Number of compositions of n^2, such that each element of [n] is used at least once as a part.

1, 1, 3, 72, 6232, 1621620, 1241237520, 2675188471920, 15634073104902000, 239929277724680059440, 9411787539302194544158080, 922671287397731617736789070720, 221805878984619105095368813189002240, 128660270226206951104782827202740054476800

Offset

0,3

Comments

Some parts can be larger than n. Adding the condition that parts cannot be larger than n, we get A332721. Removing from A332721 the condition that each element of [n] has to be used, we get A332716.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

Example

a(4) = 6232: all permutations of 4321111111, 432211111, 43222111, 4322221, 43321111, 4332211, 433321, 4432111, 443221, 543211, 64321.

Maple

b:= proc(n, i, p, m) option remember; `if`(n=0, p!,
      `if`(i<1, 0, (t-> add(b(n-i*j, i-1, p+j, t)/(j+
      `if`(t=0, 1, 0))!, j=0..n/i))(`if`(i>m, m, 0))))
    end:
a:= n-> b(n*(n-1)/2$2, n$2):
seq(a(n), n=0..15);

Mathematica

b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, p!,
     If[i < 1, 0, Function[t, [b[n - i*j, i - 1, p + j, t]/(j +
     If[t == 0, 1, 0])!, {j, 0, n/i}]][If[i > m, m, 0]]]];
a[n_] := b[n(n-1)/2, n(n-1)/2, n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 08 2021, after Alois P. Heinz *)

Crossrefs

Cf. A011782, A103488, A332716, A332721, A332796.

Sequence in context: A228712 A300967 A332721 * A202948 A213986 A156908

Adjacent sequences: A332744 A332745 A332746 * A332748 A332749 A332750

Keyword

nonn

Author

Alois P. Heinz, Feb 21 2020

Status

approved