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).
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]
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
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Aug 09 2015
STATUS
approved