A292713 Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 1055520, 10465854, 113479756, 1350508150, 17373376892, 241576630993, 3596468789967, 57232276979726, 967517444008250, 17339617861447844, 328037083000497867, 6537494747743375847, 136820214583596515519
Offset: 0
Keywords
Examples
a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 4: {aa}, {ab}, {ba}, {a,a}.
a(3) = 14: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}, {a,a,a}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
Programs
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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)): A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d* g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n) end: a:= n-> A(n$2): seq(a(n), n=0..25); -
Mathematica
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*g[d, k], {d, Divisors[j]}]* A[n - j, k], {j, 1, n}]/n]; a[n_] := A[n, n]; a /@ Range[0, 25] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)
Formula
a(n) = [x^n] Product_{j=1..n} 1/(1-x^j)^A226873(j,n).
a(n) = A292712(n,n).
a(n) ~ c * n!, where c = A247551 = 2.5294774720791526... - Vaclav Kotesovec, Oct 05 2017