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 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. 20
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..24, flattened FORMULA T(n,k) = Sum_{i=0..k} C(k,i) * A257783(n,k-i). EXAMPLE 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, ... 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 *) 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. Sequence in context: A361952 A323224 A118340 * A210391 A071921 A003992 Adjacent sequences: A213273 A213274 A213275 * A213277 A213278 A213279 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 08 2012 STATUS approved

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Last modified June 6 10:50 EDT 2023. Contains 363142 sequences. (Running on oeis4.)