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

0,19

Links

Alois P. Heinz, Antidiagonals n = 0..55, flattened

Wikipedia, Partition of a set

Formula

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

Example

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, ...

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);

Crossrefs

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

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

Main diagonal gives A370364.

Antidiagonal sums give A370365.

Cf. A060540, A370366.

Sequence in context: A176442 A361602 A281115 * A249776 A348970 A053878

Adjacent sequences: A370360 A370361 A370362 * A370364 A370365 A370366

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 16 2024

Status

approved