A260875 Square array read by ascending antidiagonals: number of m-shape complementary Bell numbers.

1, 1, -1, 1, -1, 0, 1, -1, 0, -1, 1, -1, 2, 1, 1, 1, -1, 9, -1, 1, -1, 1, -1, 34, -197, -43, -2, 1, 1, -1, 125, -5281, 6841, 254, -9, -1, 1, -1, 461, -123124, 2185429, -254801, 4157, -9, 2, 1, -1, 1715, -2840293, 465693001, -1854147586, -3000807, -70981, 50, -2

Offset: 1

Author

Peter Luschny, Aug 09 2015

Keywords

• sign
• easy
• tabl

Comments

A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n.

M-complementary Bell numbers count the m-shape set partitions which have even length minus the number of such partitions which have odd length.

If m=0 all possible sizes are zero. Thus in this case the complementary Bell numbers count the integer partitions of n into an even number of parts minus the number of integer partitions of n into an odd number of parts (A081362).

If m=1 the set is {1,2,...,n} and the complementary Bell numbers count the set partitions which have even length minus the set partitions which have odd length (A000587).

If m=2 the set is {1,2,...,2n} and the complementary Bell numbers count the set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A260884).

Examples

			[ n ] [ 0   1   2      3        4            5              6]
[ m ] --------------------------------------------------------
[ 0 ] [ 1, -1,  0,    -1,       1,          -1,             1] A081362
[ 1 ] [ 1, -1,  0,     1,       1,          -2,            -9] A000587
[ 2 ] [ 1, -1,  2,    -1,     -43,         254,          4157] A260884
[ 3 ] [ 1, -1,  9,  -197,    6841,     -254801,      -3000807]
[ 4 ] [ 1, -1, 34, -5281, 2185429, -1854147586, 2755045819549]
      A010763,
For example the number of set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 84, 280 respectively. Thus A(3,3) = -1 + 84 - 280 = -197.
Formatted as a triangle:
[1]
[1, -1]
[1, -1,   0]
[1, -1,   0,    -1]
[1, -1,   2,     1,    1]
[1, -1,   9,    -1,    1,  -1]
[1, -1,  34,  -197,  -43,  -2,  1]
[1, -1, 125, -5281, 6841, 254, -9, -1]
		

Crossrefs

Cf. A000587, A010763, A081362, A260877, A260833, A260884, A260876.

Programs

Sage

def A260875(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum((-1)^len(s)*SetPartitions(sum(s),s).cardinality() for s in shapes)
for m in (0..4): print([A260875(m,n) for n in (0..6)])