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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.
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%I #33 Sep 03 2024 17:47:14

%S 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,

%T 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,

%U 17,320,21147,0,1,1,0,0,0,1,1,8,108,1436,115975,0

%N 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.

%H Alois P. Heinz, <a href="/A375924/b375924.txt">Antidiagonals n = 0..40, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e 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.

%e 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.

%e 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.

%e A(8,5) = 1: 12345678.

%e A(8,8) = 4: 18|27|36|45, 1|278|36|45, 1|27|368|45, 1|27|36|458.

%e 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.

%e 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.

%e A(9,10) = 1: 1|29|38|47|56.

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 5, 2, 0, 1, 1, 0, 0, 0, 0, 0, ...

%e 0, 15, 4, 1, 2, 0, 0, 0, 1, 1, 0, ...

%e 0, 52, 10, 3, 3, 0, 1, 1, 0, 0, 0, ...

%e 0, 203, 28, 9, 1, 1, 3, 0, 0, 1, 1, ...

%e 0, 877, 96, 17, 8, 15, 4, 0, 1, 1, 0, ...

%e 0, 4140, 320, 108, 32, 1, 0, 1, 4, 0, 0, ...

%e 0, 21147, 1436, 324, 51, 10, 87, 72, 5, 0, 1, ...

%e 0, 115975, 5556, 1409, 621, 50, 1, 0, 0, 1, 5, ...

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

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

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

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

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

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Sep 02 2024