%I #15 Mar 04 2022 10:57:14
%S 1,1,2,5,14,42,131,425,1405,4768,16355,57112,200908,715945,2563950,
%T 9275069,33670472,123183827,451968108,1668602686,6173397451,
%U 22959862085,85535398344,320037864300,1199162682255,4509735549128,16979368985245,64134400212097
%N Number of compositions of n where the (possibly scattered) maximal subsequence of part i with multiplicity j is marked with a word of length j over an a(i-1)-ary alphabet whose letters appear in alphabetical order.
%H Alois P. Heinz, <a href="/A261894/b261894.txt">Table of n, a(n) for n = 0..1000</a>
%e a(4) = 14: 4a, 4b, 4c, 4d, 4e, 3a1a, 3b1a, 1a3a, 1a3b, 22aa 2a11aa, 12a1aa, 11aa2a, 1111aaaa.
%e a(5) = 42: 5a, 5b, 5c, 5d, 5e, 5f, 5g, 5h, 5i, 5j, 5k, 5l, 5m, 5n, 4a1a, 4b1a, 4c1a, 4d1a, 4e1a, 1a4a, 1a4b, 1a4c, 1a4d, 1a4e, 3a2a, 3b2a, 2a3a, 2a3b, 3a11aa, 3b11aa, 13a1aa, 13b1aa, 11aa3a, 11aa3b, 22aa1a, 21a2aa, 1a22aa, 2a111aaa, 12a11aaa, 112a1aaa, 111aaa2a, 11111aaaaa.
%e a(6) = 131: 6a, ..., 33aa, 33ab, 33bb, ..., 111111aaaaaa. The marked composition 33ba is not in this list.
%p b:= proc(n, i, p) option remember; `if`(n=0, p!,
%p `if`(i<1, 0, add((t-> binomial(t+j-1, t-1))(
%p a(i-1))*b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..30);
%t b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!,
%t If[i < 1, 0, Sum[Function[t, Binomial[t + j - 1, t - 1]][
%t a[i - 1]]*b[n - i*j, i - 1, p + j]/j!, {j, 0, n/i}]]];
%t a[n_] := b[n, n, 0];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 04 2022, after _Alois P. Heinz_ *)
%Y Cf. A000108 (without order restriction), A032009 (with distinct parts).
%K nonn,eigen
%O 0,3
%A _Alois P. Heinz_, Sep 05 2015