%I #9 Mar 04 2022 10:57:11

%S 1,1,3,19,115,951,10281,116313,1436499,20203795,338834053,5824666893,

%T 108142092169,2118605140237,44375797806315,1039641056342619,

%U 25413053107195539,646983321301050147,17311013062443870681,481282277347815404745,13913039361920333694165

%N 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.

%H Alois P. Heinz, <a href="/A261777/b261777.txt">Table of n, a(n) for n = 0..400</a>

%e 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.

%p with(combinat):

%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(

%p b(n-i*j, i-1, p+j)/j!*multinomial(n, n-i*j, j$i), j=0..n/i)))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..25);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t 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}]]];

%t a[n_] := b[n, n, 0];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 04 2022, after _Alois P. Heinz_ *)

%Y Cf. A000670, A261774.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 31 2015