A260833 -id:A260833 - cp's OEIS frontend

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

Peter Luschny, Aug 09 2015

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).

			[ 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]

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

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)])

A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -1, 1, -5, 1, -1, 5, -1, 21, 1, -1, 19, -61, 1, -105, 1, -1, 69, -1513, 1385, -1, 635, 1, -1, 251, -33661, 315523, -50521, 1, -4507, 1, -1, 923, -750751, 60376809, -136085041, 2702765, -1, 36457, 1, -1, 3431, -17116009, 11593285251

Offset: 1

Peter Luschny, Aug 09 2015

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. It is ordered if the positions of the blocks are taken into account.
M-shape Euler numbers count the ordered 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 m-shape Euler numbers count the ordered integer partitions of n into an even number of parts minus the number of ordered integer partitions of n into an odd number of parts (A260845).
If m=1 the set is {1,2,...,n} and the set of all possible sizes are the integer partitions of n. Thus the Euler numbers count the ordered set partitions which have even length minus the set partitions which have odd length (A033999).
If m=2 the set is {1,2,...,2n} and the 2-shape Euler numbers count the ordered set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A028296).

			[ n ] [0   1   2       3         4              5                 6]
[ m ] --------------------------------------------------------------
[ 0 ] [1, -1,  1,     -5,       21,          -105,              635] A260845
[ 1 ] [1, -1,  1,     -1,        1,            -1,                1] A033999
[ 2 ] [1, -1,  5,    -61,     1385,        -50521,          2702765] A028296
[ 3 ] [1, -1, 19,  -1513,   315523,    -136085041,     105261234643] A002115
[ 4 ] [1, -1, 69, -33661, 60376809, -288294050521, 3019098162602349] A211212
         A030662,A211213,  A181991,
For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 168, 1680 respectively. Thus A(3,3) = -1 + 168 - 1680 = -1513.
Formatted as a triangle:
[1]
[1, -1]
[1, -1,  1]
[1, -1,  1,    -5]
[1, -1,  5,    -1,   21]
[1, -1, 19,   -61,    1, -105]
[1, -1, 69, -1513, 1385,   -1, 635]

Cf. A002115, A028296, A030662, A033999, A181991, A211212, A211213, A260845, A260833, A260875, A260876.

def A260877(m,n):
    shapes = ([x*m for x in p] for p in Partitions(n).list())
    return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s). cardinality() for s in shapes)
for m in (0..5): print([A260877(m,n) for n in (0..7)])

cp's OEIS Frontend

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

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