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

1,13

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).

Links

Table of n, a(n) for n=1..55.

Example

[ 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]

Prog

(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)])

Crossrefs

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

Sequence in context: A364457 A305567 A134744 * A351350 A176250 A248975

Adjacent sequences: A260872 A260873 A260874 * A260876 A260877 A260878

Keyword

sign,easy,tabl

Author

Peter Luschny, Aug 09 2015

Status

approved