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

0,9

Comments

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).

Links

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

Example

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;
  ...

Maple

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);

Crossrefs

Columns k=0-2 give: A000007, A057427, A000225(n-1).

Row sums give A178682.

T(2n,n) gives A364322.

Cf. A364310.

Sequence in context: A227320 A318507 A055807 * A213060 A272008 A054024

Adjacent sequences: A364282 A364283 A364284 * A364286 A364287 A364288

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 17 2023

Status

approved