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Number of m-shape set partitions, square array read by ascending antidiagonals, A(m, n) for m, n >= 0.
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Irwin (1916) proved the following combinatorial result: Assume r_1, r_2, ..., r_n are positive integers and we have r_1*r_2*...*r_n objects. We divide them into r_1 classes of r_2*r_3*...*r_n objects each, then each class into r_2 sub-classes subclasses of r_3*...*r_n objects each, and so on. We call each such classification, without reference to order, a "classification" par excellence. He proved that the total number of classifications is (r_1*r_2*...*r_n)!/( r1! * (r_2!)^(r_1) * (r_3!)^(r_1*r_2) * ... (r_n!)^(r_1*r_2*...*r_{n-1}) ).
(SageSageMath)
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shapes = [map(lambda x: [x*m, for x in p) ] for p in Partitions(n).list()]
return sum([SetPartitions(sum(s), s).cardinality() for s in shapes])
for m in (0..4): print ([A260876(m, n) for n in (0..6)])
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