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%I #22 Feb 09 2021 10:59:44
%S 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,
%T 480,720,0,5,245,1106,2052,2820,3240,5040,0,6,621,3822,10576,14820,
%U 21480,25200,40320,0,8,1431,13840,41896,86724,124440,186480,221760,362880
%N 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.
%C 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.
%H Alois P. Heinz, <a href="/A319498/b319498.txt">Rows n = 0..140, flattened</a>
%F T(n,k) = A292795(n,k) - A292795(n,k-1) for k > 0, T(n,0) = A000007(n).
%e T(3,1) = 2: {aaa}, {aa,a}.
%e T(3,2) = 5: {aab}, {aba}, {baa}, {ab,a}, {ba,a}.
%e T(3,3) = 6: {abc}, {acb}, {bac}, {bca}, {cab}, {cba}.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 2, 5, 6;
%e 0, 2, 16, 18, 24;
%e 0, 3, 39, 80, 84, 120;
%e 0, 4, 106, 323, 432, 480, 720;
%e 0, 5, 245, 1106, 2052, 2820, 3240, 5040;
%e 0, 6, 621, 3822, 10576, 14820, 21480, 25200, 40320;
%e ...
%p b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
%p add(b(n-j, j, t-1)/j!, j=i..n/t))
%p end:
%p g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
%p end:
%p T:= (n, k)-> h(n$2, k) -`if`(k=0, 0, h(n$2, k-1)):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!,
%t Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
%t g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
%t h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0,
%t Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]];
%t T[n_, k_] := h[n, n, k] - If[k == 0, 0, h[n, n, k - 1]];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 09 2021, after _Alois P. Heinz_ *)
%Y Columns k=0-1 give: A000007, A000009 (for n>0).
%Y Row sums give A292796.
%Y Main diagonal gives A000142.
%Y First lower diagonal gives A038720 (for n>1).
%Y Cf. A292795, A319495.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Sep 20 2018
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