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 A319498 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that for k>0 the k-th letter occurs at least once and within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 4
 1, 0, 1, 0, 1, 2, 0, 2, 5, 6, 0, 2, 16, 18, 24, 0, 3, 39, 80, 84, 120, 0, 4, 106, 323, 432, 480, 720, 0, 5, 245, 1106, 2052, 2820, 3240, 5040, 0, 6, 621, 3822, 10576, 14820, 21480, 25200, 40320, 0, 8, 1431, 13840, 41896, 86724, 124440, 186480, 221760, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k <= n.  T(n,k) = 0 for k > n. LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,k) = A292795(n,k) - A292795(n,k-1) for k > 0, T(n,0) = A000007(n). EXAMPLE T(3,1) = 2: {aaa}, {aa,a}. T(3,2) = 5: {aab}, {aba}, {baa}, {ab,a}, {ba,a}. T(3,3) = 6: {abc}, {acb}, {bac}, {bca}, {cab}, {cba}. Triangle T(n,k) begins:   1;   0, 1;   0, 1,    2;   0, 2,    5,     6;   0, 2,   16,    18,    24;   0, 3,   39,    80,    84,   120;   0, 4,  106,   323,   432,   480,    720;   0, 5,  245,  1106,  2052,  2820,   3240,   5040;   0, 6,  621,  3822, 10576, 14820,  21480,  25200,  40320;   ... MAPLE b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,       add(b(n-j, j, t-1)/j!, j=i..n/t))     end: g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)): h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))     end: T:= (n, k)-> h(n\$2, k) -`if`(k=0, 0, h(n\$2, k-1)): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!,      Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0,      Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]]; T[n_, k_] := h[n, n, k] - If[k == 0, 0, h[n, n, k - 1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *) CROSSREFS Columns k=0-1 give: A000007, A000009 (for n>0). Row sums give A292796. Main diagonal gives A000142. First lower diagonal gives A038720 (for n>1). Cf. A292795, A319495. Sequence in context: A094721 A301951 A144529 * A011297 A110282 A024308 Adjacent sequences:  A319495 A319496 A319497 * A319499 A319500 A319501 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 20 2018 STATUS approved

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Last modified June 17 19:57 EDT 2021. Contains 345085 sequences. (Running on oeis4.)