A370367 -id:A370367 - cp's OEIS frontend

A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no 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.

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 9, 8, 0, 0, 1, 0, 34, 252, 60, 0, 0, 1, 0, 125, 5672, 14337, 544, 0, 0, 1, 0, 461, 125750, 2604732, 1327104, 6040, 0, 0, 1, 0, 1715, 2857472, 488360625, 2533087904, 182407545, 79008, 0, 0

Offset: 0

Alois P. Heinz, Feb 16 2024

			A(2,3) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Square array A(n,k) begins:
  1, 1,   1,       1,          1,             1, ...
  0, 0,   0,       0,          0,             0, ...
  0, 0,   2,       9,         34,           125, ...
  0, 0,   8,     252,       5672,        125750, ...
  0, 0,  60,   14337,    2604732,     488360625, ...
  0, 0, 544, 1327104, 2533087904, 5192229797500, ...

Alois P. Heinz, Antidiagonals n = 0..54, flattened
Wikipedia, Partition of a set

Columns k=0+1,2-3 give: A000007, A053871, A370357.
Rows n=0-2 give: A000012, A000004, A010763(n-1) for k>0.
Main diagonal gives A370367.
Antidiagonal sums give A370368.
Cf. A060540, A370363.

A:= proc(n, k) `if`(k=0,`if`(n=0, 1, 0), add(
      (-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = A060540(n,k) - A370363(n,k) for n,k >= 1.

A370364 Number of partitions of [n^2] into n sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

Original entry on oeis.org

0, 1, 1, 28, 22893, 2443061876, 68542265471953355, 833412961429901104030214430, 6514551431426932053792271970458170132097, 45458343253887079540702419310885199704811913950207054152, 375236832464739513549091449370258959406125572044428827214970469920572831639

Offset: 0

Alois P. Heinz, Feb 16 2024

nonn

			a(1) = 1: 1.
a(2) = 1: 12|34.
a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.

Alois P. Heinz, Table of n, a(n) for n = 0..27
Wikipedia, Partition of a set

Main diagonal of A370363.

Cf. A057599, A370367.

a:= n-> add((-1)^(n-j+1)*binomial(n, j)*(n*j)!/(j!*n!^j), j=0..n-1):
seq(a(n), n=0..10);

a(n) = Sum_{j=0..n-1} (-1)^(n-j+1)*binomial(n,j)*(n*j)!/(j!*n!^j).
a(n) = A370363(n,n).
a(n) = A057599(n) - A370367(n).

cp's OEIS Frontend

A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no 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.

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