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%I #28 Mar 06 2020 03:12:58
%S 1,1,-1,1,-1,1,1,-1,1,-5,1,-1,5,-1,21,1,-1,19,-61,1,-105,1,-1,69,
%T -1513,1385,-1,635,1,-1,251,-33661,315523,-50521,1,-4507,1,-1,923,
%U -750751,60376809,-136085041,2702765,-1,36457,1,-1,3431,-17116009,11593285251
%N Square array read by ascending antidiagonals: number of m-shape Euler 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. It is ordered if the positions of the blocks are taken into account.
%C 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.
%C 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).
%C 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).
%C 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).
%e [ n ] [0 1 2 3 4 5 6]
%e [ m ] --------------------------------------------------------------
%e [ 0 ] [1, -1, 1, -5, 21, -105, 635] A260845
%e [ 1 ] [1, -1, 1, -1, 1, -1, 1] A033999
%e [ 2 ] [1, -1, 5, -61, 1385, -50521, 2702765] A028296
%e [ 3 ] [1, -1, 19, -1513, 315523, -136085041, 105261234643] A002115
%e [ 4 ] [1, -1, 69, -33661, 60376809, -288294050521, 3019098162602349] A211212
%e A030662,A211213, A181991,
%e 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.
%e Formatted as a triangle:
%e [1]
%e [1, -1]
%e [1, -1, 1]
%e [1, -1, 1, -5]
%e [1, -1, 5, -1, 21]
%e [1, -1, 19, -61, 1, -105]
%e [1, -1, 69, -1513, 1385, -1, 635]
%o (Sage)
%o def A260877(m,n):
%o shapes = ([x*m for x in p] for p in Partitions(n).list())
%o return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s). cardinality() for s in shapes)
%o for m in (0..5): print([A260877(m,n) for n in (0..7)])
%Y Cf. A002115, A028296, A030662, A033999, A181991, A211212, A211213, A260845, A260833, A260875, A260876.
%K sign,tabl
%O 1,10
%A _Peter Luschny_, Aug 09 2015
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