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Number of sets of nonempty words with a total of n letters over ternary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
2

%I #6 Jan 06 2021 19:43:32

%S 1,1,3,13,36,122,433,1356,4449,15279,48567,158837,532415,1704777,

%T 5547148,18335536,58815602,190574866,623885902,2000945191,6459510350,

%U 20998728429,67275468661,216477522426,699952967976,2239210854373,7184690267832,23131348476391

%N Number of sets of nonempty words with a total of n letters over ternary 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="/A340410/b340410.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{j>=1} (1+x^j)^A092255(j).

%e a(3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {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, 3)):

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

%Y Column k=3 of A292795.

%Y Cf. A092255.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 06 2021