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A257783
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Number T(n,k) of words w of length n such that each letter of the k-ary alphabet is used at least once and 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; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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11
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1, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 7, 12, 24, 0, 1, 12, 35, 60, 120, 0, 1, 25, 87, 210, 360, 720, 0, 1, 44, 232, 609, 1470, 2520, 5040, 0, 1, 89, 599, 1961, 4872, 11760, 20160, 40320, 0, 1, 160, 1591, 5952, 17649, 43848, 105840, 181440, 362880, 0, 1, 321, 4202, 19255, 60465, 176490, 438480, 1058400, 1814400, 3628800
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OFFSET
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0,6
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COMMENTS
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Row n is the inverse binomial transform of the n-th row of array A213276.
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LINKS
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Alois P. Heinz, Rows n = 0..20, flattened
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A213276(n,k-i).
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EXAMPLE
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T(5,2) = 12: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, baaaa, baaab, baaba.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 3, 6;
0, 1, 7, 12, 24;
0, 1, 12, 35, 60, 120;
0, 1, 25, 87, 210, 360, 720;
0, 1, 44, 232, 609, 1470, 2520, 5040;
0, 1, 89, 599, 1961, 4872, 11760, 20160, 40320;
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MATHEMATICA
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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];
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]}]];
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]]]];
A[n_, k_] := h[n, k, 0, {}];
T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*A[n, k - i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A213276 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A057427, A321838, A321839, A321840, A321841, A321842, A321843, A321844, A321845, A321846.
Main diagonal gives A000142.
T(n+1,n) = A001710(n+1) (for n>0).
Cf. A213276.
Sequence in context: A195772 A330618 A062104 * A226874 A267901 A276561
Adjacent sequences: A257780 A257781 A257782 * A257784 A257785 A257786
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, May 08 2015
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STATUS
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approved
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