A336032 Number of compositions of n such that the set of parts and the set of multiplicities of parts are disjoint.

1, 0, 2, 2, 2, 4, 6, 6, 14, 34, 79, 159, 227, 429, 727, 1146, 1999, 3238, 5018, 8976, 14977, 24768, 38400, 70678, 152535, 295493, 617675, 1404099, 3023086, 6685876, 14230031, 30218806, 62175519, 127820798, 257285277, 516574751, 1021334631, 2009999405, 3917878730

Offset: 0

Author

Alois P. Heinz, Jul 07 2020

Keywords

• nonn

Crossrefs

Cf. A114639, A336031.

Programs

Maple

b:= proc(n, i, p, f, g) option remember; `if`(f intersect g<>{}, 0,
      `if`(n=0, p!, `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0,
       f, {f[], i}), `if`(j=0, g, {g[], j}))/j!, j=0..n/i))))
    end:
a:= n-> b(n$2, 0, {}$2):
seq(a(n), n=0..32);

Mathematica

b[n_, i_, p_, f_, g_] := b[n, i, p, f, g] = If[f ~Intersection~ g != {}, 0,
     If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j,
     If[j == 0, f, Union@Append[f, i]],
     If[j == 0, g, Union@Append[g, j]]]/j!, {j, 0, n/i}]]]];
a[n_] := b[n, n, 0, {}, {}];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)