A375924 Number A(n,k) of partitions of [n] such that the element sum of each block is one more than a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 5, 0, 1, 1, 0, 2, 15, 0, 1, 1, 0, 0, 4, 52, 0, 1, 1, 0, 1, 1, 10, 203, 0, 1, 1, 0, 1, 2, 3, 28, 877, 0, 1, 1, 0, 0, 0, 3, 9, 96, 4140, 0, 1, 1, 0, 0, 0, 0, 1, 17, 320, 21147, 0, 1, 1, 0, 0, 0, 1, 1, 8, 108, 1436, 115975, 0

Offset: 0

Author

Alois P. Heinz, Sep 02 2024

Keywords

• nonn
• tabl

Examples

			A(5,2) = 10: 12345, 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34.
A(6,3) = 9: 136|25|4, 13|256|4, 13|25|46, 16|235|4, 1|2356|4, 1|235|46, 16|25|34, 1|256|34, 1|25|346.
A(7,4) = 8: 14|23|5|67, 1|234|5|67, 1|23|45|67, 1|23|467|5, 14|27|36|5, 1|247|36|5, 1|27|346|5, 1|27|36|45.
A(8,5) = 1: 12345678.
A(8,8) = 4: 18|27|36|45, 1|278|36|45, 1|27|368|45, 1|27|36|458.
A(9,6) = 87: 123469|58|7, 12349|568|7, 12349|58|67, 123568|49|7, ..., 1|25|346789, 16|289|3457, 1|2689|3457, 1|289|34567.
A(9,8) = 5: 18|27|36|45|9, 1|278|36|45|9, 1|27|368|45|9, 1|27|36|458|9, 1|27|36|45|89.
A(9,10) = 1: 1|29|38|47|56.
Square array A(n,k) begins:
  1,      1,    1,    1,   1,  1,  1,  1, 1, 1, 1, ...
  1,      1,    1,    1,   1,  1,  1,  1, 1, 1, 1, ...
  0,      2,    1,    0,   0,  0,  0,  0, 0, 0, 0, ...
  0,      5,    2,    0,   1,  1,  0,  0, 0, 0, 0, ...
  0,     15,    4,    1,   2,  0,  0,  0, 1, 1, 0, ...
  0,     52,   10,    3,   3,  0,  1,  1, 0, 0, 0, ...
  0,    203,   28,    9,   1,  1,  3,  0, 0, 1, 1, ...
  0,    877,   96,   17,   8, 15,  4,  0, 1, 1, 0, ...
  0,   4140,  320,  108,  32,  1,  0,  1, 4, 0, 0, ...
  0,  21147, 1436,  324,  51, 10, 87, 72, 5, 0, 1, ...
  0, 115975, 5556, 1409, 621, 50,  1,  0, 0, 1, 5, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..40, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-10 give: A019590(n+1), A000110, A369079, A374692, A375939, A375940, A375941, A375942, A375943, A375944, A375957.

Rows n=1-2 give: A000012, A033322 (for k>=1).

Main diagonal gives A142150 (for n>=2).

A(n+1,n) gives A158416 (for n>=2).

A(n,n+1) gives A135528(n+1).