login
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.
6

%I #21 Feb 19 2024 18:28:37

%S 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,

%T 1063,401,1,1,0,1,1,376,22893,74296,4355,1,1,0,1,1,1384,503751,

%U 13080721,8182855,56127,1,1,0,1,1,5146,11432655,2443061876,15237712355,1305232804,836353,1,1

%N 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.

%H Alois P. Heinz, <a href="/A370363/b370363.txt">Antidiagonals n = 0..55, flattened</a>

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

%F A(n,k) = A060540(n,k) - A370366(n,k) for n,k >= 1.

%e 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.

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

%e 0, 0, 0, 0, 0, 0, ...

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

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

%e 1, 1, 7, 28, 103, 376, ...

%e 1, 1, 45, 1063, 22893, 503751, ...

%e 1, 1, 401, 74296, 13080721, 2443061876, ...

%p A:= proc(n, k) option remember; `if`(k=0, signum(n), add(

%p (-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1))

%p end:

%p seq(seq(A(n, d-n), n=0..d), d=0..10);

%Y Columns k=0+1,2-3 give: A057427, A370253, A370358.

%Y Rows n=0,1+2,3 give: A000004, A000012, A370487.

%Y Main diagonal gives A370364.

%Y Antidiagonal sums give A370365.

%Y Cf. A060540, A370366.

%K nonn,tabl

%O 0,19

%A _Alois P. Heinz_, Feb 16 2024