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, 1, 4, 18, 118, 895, 8151, 83916, 977026, 12602451, 178880725, 2766415036, 46314488705, 834067614601, 16074694453741, 330017679352180, 7188779521480810

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

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

Example

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

Maple

h:= proc(n, k, m, l) option remember;
      `if`(n=0 and k=0, b(l), `if`(k=0 or n>0 and n<m, 0,
       add(h(n-j, k-1, max(m, j), [j, l[]]), j=max(1, m)..n)
          +h(n, k-1, m, [0, l[]], [])))
    end:
b:= proc(l) option remember;
      `if`({l[]} minus {0}={}, 1, add(`if`(g(l, i),
       b(subsop(i=l[i]-1, l)), 0), i=1..nops(l)))
    end:
g:= proc(l, i) local j;
      if l[i]<1     then return false
    elif l[i]>1     then for j from i+1 to nops(l) do
      if l[i]<=l[j] then return false
    elif l[j]>0     then break
      fi od fi; true
    end:
a:= n-> h(n$2, 0, []):
seq(a(n), n=0..10);  # Alois P. Heinz, Mar 29 2020

Mathematica

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]]]];
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]}]];
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];
a[n_] := h[n, n, 0, {}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)

Crossrefs

Main diagonal of A213276.

Sequence in context: A278994 A223008 A162224 * A296982 A222375 A053529

Adjacent sequences: A321701 A321702 A321703 * A321705 A321706 A321707

Keyword

nonn,more

Author

Alois P. Heinz, Nov 17 2018

Status

approved