A340409 Number of sets of nonempty words with a total of n letters over binary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

1, 1, 3, 7, 18, 42, 110, 250, 627, 1439, 3523, 8063, 19374, 44274, 104816, 238976, 559171, 1271295, 2946901, 6679741, 15363719, 34719631, 79335385, 178749829, 406164359, 912475815, 2063298409, 4622461673, 10407679805, 23254807241, 52160338735, 116252939071

Offset: 0

Author

Alois P. Heinz, Jan 06 2021

Keywords

• nonn

Examples

			a(3) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.
		

Links

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

Crossrefs

Column k=2 of A292795.

Cf. A027306.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
    end:
a:= n-> h(n$2, min(n, 2)):
seq(a(n), n=0..32);

Formula

G.f.: Product_{j>=1} (1+x^j)^A027306(j).