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A364310 Number T(n,k) of partitions of n into k parts where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A364310 #18 Nov 18 2023 07:27:25
%S A364310 1,0,1,0,1,1,0,1,3,1,0,1,5,6,1,0,1,15,15,10,1,0,1,22,76,35,15,1,0,1,
%T A364310 63,168,252,70,21,1,0,1,93,574,785,658,126,28,1,0,1,255,2188,3066,
%U A364310 2739,1470,210,36,1,0,1,386,5490,18235,12181,7857,2940,330,45,1
%N A364310 Number T(n,k) of partitions of n into k parts where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A364310 Alois P. Heinz, <a href="/A364310/b364310.txt">Rows n = 0..150, flattened</a>
%e A364310 T(4,1) = 1: 4abcd.
%e A364310 T(4,2) = 5: 3abc1d, 3abd1c, 3acd1b, 3bcd1a, 22abcd.
%e A364310 T(4,3) = 6: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab.
%e A364310 T(4,4) = 1: 1111abcd.
%e A364310 Triangle T(n,k) begins:
%e A364310   1;
%e A364310   0, 1;
%e A364310   0, 1,   1;
%e A364310   0, 1,   3,    1;
%e A364310   0, 1,   5,    6,    1;
%e A364310   0, 1,  15,   15,   10,    1;
%e A364310   0, 1,  22,   76,   35,   15,    1;
%e A364310   0, 1,  63,  168,  252,   70,   21,   1;
%e A364310   0, 1,  93,  574,  785,  658,  126,  28,  1;
%e A364310   0, 1, 255, 2188, 3066, 2739, 1470, 210, 36, 1;
%e A364310   ...
%p A364310 b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p A364310       add(b(n-i*j, i-1)*x^j*binomial(n, i*j), j=0..n/i))))
%p A364310     end:
%p A364310 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A364310 seq(T(n), n=0..12);
%t A364310 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
%t A364310    Sum[b[n - i*j, i - 1]*x^j*Binomial[n, i*j], {j, 0, n/i}]]]];
%t A364310 T[n_] := CoefficientList[b[n, n], x];
%t A364310 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 18 2023, after _Alois P. Heinz_ *)
%Y A364310 Columns k=0-1 give: A000007, A057427.
%Y A364310 Row sums give A178682.
%Y A364310 T(n,n) gives A000012.
%Y A364310 T(n+1,n) gives A000217.
%Y A364310 T(n+2,n) gives A000332(n+3).
%Y A364310 Cf. A364285.
%K A364310 nonn,tabl
%O A364310 0,9
%A A364310 _Alois P. Heinz_, Jul 18 2023