A213276 Number A(n,k) of n-length words w over a k-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; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 5, 1, 0, 1, 5, 16, 18, 9, 1, 0, 1, 6, 25, 46, 36, 14, 1, 0, 1, 7, 36, 95, 118, 74, 27, 1, 0, 1, 8, 49, 171, 315, 276, 165, 46, 1, 0, 1, 9, 64, 280, 711, 895, 712, 367, 91, 1, 0, 1, 10, 81, 428, 1414, 2506, 2535, 1805, 869, 162, 1, 0

Offset: 0

Author

Alois P. Heinz, Jun 08 2012

Keywords

• nonn
• tabl

Examples

			A(0,k) = 1: the empty word.
A(n,1) = 1: (a)^n for alphabet {a}.
A(1,k) = k: number of words = size of the alphabet.
A(2,k) = k^2: all words with 2 letters from the alphabet.
A(3,2) = 5: aaa, aab, aba, baa, bbb for alphabet {a,b}.
A(3,3) = 18: aaa, aab, aac, aba, abc, aca, acb, baa, bac, bbb, bbc, bca, bcb, caa, cab, cba, cbb, ccc.
A(4,2) = 9: aaaa, aaab, aaba, aabb, abaa, abab, baaa, baab, bbbb.
A(5,2) = 14: aaaaa, aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, baaaa, baaab, baaba, bbbbb.
Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,     1,     1, ...
  0, 1,  2,   3,    4,    5,     6,     7, ...
  0, 1,  4,   9,   16,   25,    36,    49, ...
  0, 1,  5,  18,   46,   95,   171,   280, ...
  0, 1,  9,  36,  118,  315,   711,  1414, ...
  0, 1, 14,  74,  276,  895,  2506,  6104, ...
  0, 1, 27, 165,  712, 2535,  8151, 23527, ...
  0, 1, 46, 367, 1805, 7280, 25781, 83916, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..24, flattened

Crossrefs

Columns k=0-10 give: A000007, A000012, A213290, A213291, A213292, A213293, A213294, A213295, A213296, A213297, A213298.

Rows n=0-10 give: A000012, A001477, A000290, A081435(k-1), A213283, A213284, A213285, A213286, A213287, A213288, A213289.

Main diagonal gives A321704.

Cf. A182172, A213275, A257783.

Programs

Maple

A:= (n, k)-> h(n, k, 0, []):
h:= proc(n, k, m, l) option remember;
      `if`(n=0 and k=0, b(l), `if`(k=0 or n>0 and n1     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:
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

a[n_, k_] := h[n, k, 0, {}];
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];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

Formula

T(n,k) = Sum_{i=0..k} C(k,i) * A257783(n,k-i).