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A319495 Number T(n,k) of multisets 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.

%I A319495 #27 Feb 09 2021 10:59:38
%S A319495 1,0,1,0,2,2,0,3,5,6,0,5,20,18,24,0,7,46,86,84,120,0,11,137,347,456,
%T A319495 480,720,0,15,313,1216,2136,2940,3240,5040,0,22,836,4253,11128,15300,
%U A319495 22200,25200,40320,0,30,1908,15410,44308,90024,127680,191520,221760,362880
%N A319495 Number T(n,k) of multisets 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 A319495 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 A319495 Alois P. Heinz, <a href="/A319495/b319495.txt">Rows n = 0..140, flattened</a>
%F A319495 T(n,k) = A292712(n,k) - A292712(n,k-1) for k > 0, T(n,0) = A000007(n).
%e A319495 T(3,1) = 3: {aaa}, {aa,a}, {a,a,a}.
%e A319495 T(3,2) = 5: {aab}, {aba}, {baa}, {ab,a}, {ba,a}.
%e A319495 T(3,3) = 6: {abc}, {acb}, {bac}, {bca}, {cab}, {cba}.
%e A319495 Triangle T(n,k) begins:
%e A319495   1;
%e A319495   0,  1;
%e A319495   0,  2,   2;
%e A319495   0,  3,   5,    6;
%e A319495   0,  5,  20,   18,    24;
%e A319495   0,  7,  46,   86,    84,   120;
%e A319495   0, 11, 137,  347,   456,   480,   720;
%e A319495   0, 15, 313, 1216,  2136,  2940,  3240,  5040;
%e A319495   0, 22, 836, 4253, 11128, 15300, 22200, 25200, 40320;
%e A319495   ...
%p A319495 b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
%p A319495       add(b(n-j, j, t-1)/j!, j=i..n/t))
%p A319495     end:
%p A319495 g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p A319495 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
%p A319495       g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
%p A319495     end:
%p A319495 T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
%p A319495 seq(seq(T(n, k), k=0..n), n=0..12);
%t A319495 b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!,
%t A319495      Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
%t A319495 g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
%t A319495 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*
%t A319495      g[d, k], {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
%t A319495 T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t A319495 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 09 2021, after _Alois P. Heinz_ *)
%Y A319495 Columns k=0-1 give: A000007, A000041 (for n>0).
%Y A319495 Row sums give A292713.
%Y A319495 Main diagonal gives A000142.
%Y A319495 First lower diagonal gives A038720.
%Y A319495 Cf. A292712, A319498.
%K A319495 nonn,tabl
%O A319495 0,5
%A A319495 _Alois P. Heinz_, Sep 20 2018

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