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Square array read by ascending antidiagonals: number of m-shape complementary Bell numbers.
4

%I #27 Mar 16 2020 08:51:35

%S 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,

%T -2,1,1,-1,125,-5281,6841,254,-9,-1,1,-1,461,-123124,2185429,-254801,

%U 4157,-9,2,1,-1,1715,-2840293,465693001,-1854147586,-3000807,-70981,50,-2

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

%C 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.

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

%C 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).

%C 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).

%C 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).

%e [ n ] [ 0 1 2 3 4 5 6]

%e [ m ] --------------------------------------------------------

%e [ 0 ] [ 1, -1, 0, -1, 1, -1, 1] A081362

%e [ 1 ] [ 1, -1, 0, 1, 1, -2, -9] A000587

%e [ 2 ] [ 1, -1, 2, -1, -43, 254, 4157] A260884

%e [ 3 ] [ 1, -1, 9, -197, 6841, -254801, -3000807]

%e [ 4 ] [ 1, -1, 34, -5281, 2185429, -1854147586, 2755045819549]

%e A010763,

%e 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.

%e Formatted as a triangle:

%e [1]

%e [1, -1]

%e [1, -1, 0]

%e [1, -1, 0, -1]

%e [1, -1, 2, 1, 1]

%e [1, -1, 9, -1, 1, -1]

%e [1, -1, 34, -197, -43, -2, 1]

%e [1, -1, 125, -5281, 6841, 254, -9, -1]

%o (Sage)

%o def A260875(m, n):

%o shapes = ([x*m for x in p] for p in Partitions(n))

%o return sum((-1)^len(s)*SetPartitions(sum(s),s).cardinality() for s in shapes)

%o for m in (0..4): print([A260875(m,n) for n in (0..6)])

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

%K sign,easy,tabl

%O 1,13

%A _Peter Luschny_, Aug 09 2015