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A321704 Number of words w of length n over an n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z. 1

%I #14 Jun 01 2022 14:44:55

%S 1,1,4,18,118,895,8151,83916,977026,12602451,178880725,2766415036,

%T 46314488705,834067614601,16074694453741,330017679352180,

%U 7188779521480810

%N Number of words w of length n over an n-ary alphabet such that for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

%F a(n) = A213276(n,n).

%e a(3) = 18: aaa, aab, aac, aba, abc, aca, acb, baa, bac, bbb, bbc, bca, bcb, caa, cab, cba, cbb, ccc.

%p h:= proc(n, k, m, l) option remember;

%p `if`(n=0 and k=0, b(l), `if`(k=0 or n>0 and n<m, 0,

%p add(h(n-j, k-1, max(m, j), [j, l[]]), j=max(1, m)..n)

%p +h(n, k-1, m, [0, l[]], [])))

%p end:

%p b:= proc(l) option remember;

%p `if`({l[]} minus {0}={}, 1, add(`if`(g(l, i),

%p b(subsop(i=l[i]-1, l)), 0), i=1..nops(l)))

%p end:

%p g:= proc(l, i) local j;

%p if l[i]<1 then return false

%p elif l[i]>1 then for j from i+1 to nops(l) do

%p if l[i]<=l[j] then return false

%p elif l[j]>0 then break

%p fi od fi; true

%p end:

%p a:= n-> h(n$2, 0, []):

%p seq(a(n), n=0..10); # _Alois P. Heinz_, Mar 29 2020

%t h[n_, k_, m_, l_] := h[n, k, m, l] = If[n == 0 && k === 0, b[l], If[k == 0 || n > 0 && n < m, 0, Sum[h[n - j, k - 1, Max[m, j], Join[{j}, l]], {j, Max[1, m], n}] + h[n, k - 1, m, Join[{0}, l]]]];

%t b[l_] := b[l] = If[Complement[l, {0}] == {}, 1, Sum[If[g[l, i], b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, Length[l]}]];

%t g[l_, i_] := Module[{j}, If[l[[i]] < 1, Return[False], If[l[[i]] > 1, For[j = i + 1, j <= Length[l], j++, If[l[[i]] <= l[[j]], Return[False], If[l[[j]] > 0, Break[]]]]]]; True];

%t a[n_] := h[n, n, 0, {}];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* _Jean-Fran├žois Alcover_, Jun 01 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A213276.

%K nonn,more

%O 0,3

%A _Alois P. Heinz_, Nov 17 2018

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Last modified June 9 09:54 EDT 2023. Contains 363178 sequences. (Running on oeis4.)