A261777 Number of compositions of n where the (possibly scattered) maximal subsequence of part i with multiplicity j is marked with i words of length j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the composition.

1, 1, 3, 19, 115, 951, 10281, 116313, 1436499, 20203795, 338834053, 5824666893, 108142092169, 2118605140237, 44375797806315, 1039641056342619, 25413053107195539, 646983321301050147, 17311013062443870681, 481282277347815404745, 13913039361920333694165

Offset: 0

Author

Alois P. Heinz, Aug 31 2015

Keywords

• nonn

Examples

			a(3) = 19: 3a|b|c, 3a|c|b, 3b|a|c, 3b|c|a, 3c|a|b, 3c|b|a, 2a|b1c, 2b|a1c, 2a|c1b, 2c|a1b, 2b|c1a, 2c|b1a, 1a2b|c, 1a2c|b, 1b2a|c, 1b2c|a, 1c2a|b, 1c2b|a, 111abc.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000670, A261774.

Programs

Maple

with(combinat):
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, i-1, p+j)/j!*multinomial(n, n-i*j, j$i), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, i-1, p + j]/j!*multinomial[n, Join[{n - i*j}, Table[j, {i}]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)