A364322 -id:A364322 - cp's OEIS frontend

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.

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, 287, 175, 42, 7, 1, 0, 1, 127, 952, 841, 280, 56, 8, 1, 0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1, 0, 1, 511, 9370, 23820, 10333, 2730, 600, 90, 10, 1

Offset: 0

Alois P. Heinz, Jul 17 2023

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.
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.
  T(4,2) = 7: (2211), (2121), (2112), (1221), (1212), (1122), (2222).
  T(5,3) = 20: (33322), (33232), (33223), (32332), (32323), (32233), (23332), (23323), (23233), (22333), (33311), (33131), (33113), (31331), (31313), (31133), (13331), (13313), (13133), (11333).

			T(4,1) = 1: 1111abcd.
T(4,2) = 7: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab, 22abcd.
T(4,3) = 4: 3abc1d, 3abd1c, 3acd1b, 3bcd1a.
T(4,4) = 1: 4abcd.
Triangle T(n,k) begins:
  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,  287,  175,   42,   7,  1;
  0, 1, 127,  952,  841,  280,  56,  8, 1;
  0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Columns k=0-2 give: A000007, A057427, A000225(n-1).
Row sums give A178682.
T(2n,n) gives A364322.
Cf. A364310.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
    end:
T:= (n, k)-> b(n, k)-`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

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