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

Author

Alois P. Heinz, Feb 21 2020

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

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 *)