A261894 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.

1, 1, 2, 5, 14, 42, 131, 425, 1405, 4768, 16355, 57112, 200908, 715945, 2563950, 9275069, 33670472, 123183827, 451968108, 1668602686, 6173397451, 22959862085, 85535398344, 320037864300, 1199162682255, 4509735549128, 16979368985245, 64134400212097

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Example

a(4) = 14: 4a, 4b, 4c, 4d, 4e, 3a1a, 3b1a, 1a3a, 1a3b, 22aa 2a11aa, 12a1aa, 11aa2a, 1111aaaa.
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.
a(6) = 131: 6a, ..., 33aa, 33ab, 33bb, ..., 111111aaaaaa. The marked composition 33ba is not in this list.

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add((t-> binomial(t+j-1, t-1))(
       a(i-1))*b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!,
     If[i < 1, 0, Sum[Function[t, Binomial[t + j - 1, t - 1]][
     a[i - 1]]*b[n - i*j, i - 1, p + j]/j!, {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000108 (without order restriction), A032009 (with distinct parts).

Sequence in context: A261588 A006930 A036767 * A287969 A061922 A293498

Adjacent sequences: A261891 A261892 A261893 * A261895 A261896 A261897

Keyword

nonn,eigen

Author

Alois P. Heinz, Sep 05 2015

Status

approved