A370363 Number A(n,k) of partitions of [k*n] into n sets of size k having at least one set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 7, 1, 1, 0, 1, 1, 28, 45, 1, 1, 0, 1, 1, 103, 1063, 401, 1, 1, 0, 1, 1, 376, 22893, 74296, 4355, 1, 1, 0, 1, 1, 1384, 503751, 13080721, 8182855, 56127, 1, 1, 0, 1, 1, 5146, 11432655, 2443061876, 15237712355, 1305232804, 836353, 1, 1
Offset: 0
Examples
A(3,2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34. Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, ... 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 1, 7, 28, 103, 376, ... 1, 1, 45, 1063, 22893, 503751, ... 1, 1, 401, 74296, 13080721, 2443061876, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..55, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
A:= proc(n, k) option remember; `if`(k=0, signum(n), add( (-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1)) end: seq(seq(A(n, d-n), n=0..d), d=0..10);