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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.
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%I #7 Jan 06 2021 19:22:04

%S 1,1,3,7,18,42,110,250,627,1439,3523,8063,19374,44274,104816,238976,

%T 559171,1271295,2946901,6679741,15363719,34719631,79335385,178749829,

%U 406164359,912475815,2063298409,4622461673,10407679805,23254807241,52160338735,116252939071

%N 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.

%H Alois P. Heinz, <a href="/A340409/b340409.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%p b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,

%p add(b(n-j, j, t-1)/j!, j=i..n/t))

%p end:

%p g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):

%p h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))

%p end:

%p a:= n-> h(n$2, min(n, 2)):

%p seq(a(n), n=0..32);

%Y Column k=2 of A292795.

%Y Cf. A027306.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 06 2021