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A364285 Number T(n,k) of partitions of n with largest part k 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 A364285 #24 Jul 20 2023 09:43:38
%S A364285 1,0,1,0,1,1,0,1,3,1,0,1,7,4,1,0,1,15,20,5,1,0,1,31,81,30,6,1,0,1,63,
%T A364285 287,175,42,7,1,0,1,127,952,841,280,56,8,1,0,1,255,3025,4545,1638,420,
%U A364285 72,9,1,0,1,511,9370,23820,10333,2730,600,90,10,1
%N A364285 Number T(n,k) of partitions of n with largest part k 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.
%C A364285 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.
%C A364285 T(n,k) is also the number of endofunctions on [n] such that k is the range maximum and the number of elements that are mapped to m is divisible by m.
%C A364285   T(4,2) = 7: (2211), (2121), (2112), (1221), (1212), (1122), (2222).
%C A364285   T(5,3) = 20: (33322), (33232), (33223), (32332), (32323), (32233), (23332), (23323), (23233), (22333), (33311), (33131), (33113), (31331), (31313), (31133), (13331), (13313), (13133), (11333).
%H A364285 Alois P. Heinz, <a href="/A364285/b364285.txt">Rows n = 0..150, flattened</a>
%e A364285 T(4,1) = 1: 1111abcd.
%e A364285 T(4,2) = 7: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab, 22abcd.
%e A364285 T(4,3) = 4: 3abc1d, 3abd1c, 3acd1b, 3bcd1a.
%e A364285 T(4,4) = 1: 4abcd.
%e A364285 Triangle T(n,k) begins:
%e A364285   1;
%e A364285   0, 1;
%e A364285   0, 1,   1;
%e A364285   0, 1,   3,    1;
%e A364285   0, 1,   7,    4,    1;
%e A364285   0, 1,  15,   20,    5,    1;
%e A364285   0, 1,  31,   81,   30,    6,   1;
%e A364285   0, 1,  63,  287,  175,   42,   7,  1;
%e A364285   0, 1, 127,  952,  841,  280,  56,  8, 1;
%e A364285   0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1;
%e A364285   ...
%p A364285 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A364285       add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
%p A364285     end:
%p A364285 T:= (n, k)-> b(n, k)-`if`(k=0, 0, b(n, k-1)):
%p A364285 seq(seq(T(n, k), k=0..n), n=0..12);
%Y A364285 Columns k=0-2 give: A000007, A057427, A000225(n-1).
%Y A364285 Row sums give A178682.
%Y A364285 T(2n,n) gives A364322.
%Y A364285 Cf. A364310.
%K A364285 nonn,tabl
%O A364285 0,9
%A A364285 _Alois P. Heinz_, Jul 17 2023