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A293113 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a 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 A293113 #27 Jul 14 2024 13:49:29
%S A293113 1,0,1,0,1,1,0,2,3,1,0,2,8,4,1,0,3,20,16,5,1,0,4,47,53,25,6,1,0,5,106,
%T A293113 173,102,36,7,1,0,6,237,532,410,172,49,8,1,0,8,522,1615,1545,813,268,
%U A293113 64,9,1,0,10,1146,4785,5784,3576,1448,394,81,10,1
%N A293113 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a 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.
%H A293113 Alois P. Heinz, <a href="/A293113/b293113.txt">Rows n = 0..40, flattened</a>
%F A293113 T(n,k) = A293112(n,k) - A293112(n,k-1) for k>0, T(n,0) = A293112(n,0).
%e A293113 Triangle T(n,k) begins:
%e A293113   1;
%e A293113   0, 1;
%e A293113   0, 1,   1;
%e A293113   0, 2,   3,   1;
%e A293113   0, 2,   8,   4,   1;
%e A293113   0, 3,  20,  16,   5,   1;
%e A293113   0, 4,  47,  53,  25,   6,  1;
%e A293113   0, 5, 106, 173, 102,  36,  7, 1;
%e A293113   0, 6, 237, 532, 410, 172, 49, 8, 1;
%e A293113   ...
%p A293113 h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
%p A293113     <j, 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
%p A293113 g:= proc(n, i, l) option remember;
%p A293113       `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
%p A293113         g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p A293113     end:
%p A293113 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293113       add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))
%p A293113     end:
%p A293113 T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
%p A293113 seq(seq(T(n, k), k=0..n), n=0..14);
%t A293113 h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]]<j, 0, 1], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]][ Length[l]];
%t A293113 g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];
%t A293113 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
%t A293113 T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
%t A293113 Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)
%Y A293113 Columns k=0-10 give: A000007, A000009 (for n>0), A293883, A293884, A293885, A293886, A293887, A293888, A293889, A293890, A293891.
%Y A293113 Row sums give A293114.
%Y A293113 T(2n,n) gives A293115.
%Y A293113 Cf. A182172, A293109, A293112.
%K A293113 nonn,tabl
%O A293113 0,8
%A A293113 _Alois P. Heinz_, Sep 30 2017