A375942 -id:A375942

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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.

+10
10

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

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OFFSET

0,9

LINKS

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Wikipedia, Partition of a set

EXAMPLE

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, ...

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, ...

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).

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 02 2024

STATUS

approved

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