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%I #39 Mar 16 2020 08:51:50
%S 1,1,1,1,1,3,1,1,3,9,1,1,7,13,35,1,1,21,121,75,161,1,1,71,1849,3907,
%T 541,913,1,1,253,35641,426405,202741,4683,6103,1,1,925,762763,
%U 65782211,203374081,15430207,47293,47319,1,1,3433,17190265,11872636325,323213457781,173959321557
%N Number of m-shape ordered set partitions, square array read by ascending antidiagonals, A(m, n) for m, n >= 0.
%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 If m = 0, all possible sizes are zero. Thus the number of ordered set partitions of 0-shape is the number of ordered partitions of n (partition numbers A101880).
%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 number of ordered set partitions of 1-shape is a Fubini number (sequence A000670).
%C If m = 2, the set is {1, 2, ..., 2n} and the number of ordered set partitions of 2-shape is also the number of 2-packed words of degree n (sequence A094088).
%F From _Petros Hadjicostas_, Aug 02 2019: (Start)
%F Conjecture: For n >= 0, let P be the set of all possible lists (a_1, ..., a_n) of nonnegative integers such that a_1*1 + a_2*2 + ... + a_n*n = n. Consider terms of the form multinomial(n*m, m*[1,..., 1, 2,..., 2,..., n,..., n]) * multinomial(a_1 + ... + a_n, [a_1,..., a_n]), where in the list [1,..., 1, 2,..., 2,..., n,..., n] the number 1 occurs a_1 times, 2 occurs a_2 times, ..., and n occurs a_n times. (Here a_n = 0 or 1.) Summing these terms over P we get A(m, n) provided m >= 1. (End)
%e [ n ] [0 1 2 3 4 5 6]
%e [ m ] -----------------------------------------------------------
%e [ 0 ] [1, 1, 3, 9, 35, 161, 913] A101880
%e [ 1 ] [1, 1, 3, 13, 75, 541, 4683] A000670
%e [ 2 ] [1, 1, 7, 121, 3907, 202741, 15430207] A094088
%e [ 3 ] [1, 1, 21, 1849, 426405, 203374081, 173959321557] A243664
%e [ 4 ] [1, 1, 71, 35641, 65782211, 323213457781, 3482943541940351] A243665
%e A244174
%e For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] is 1, 168 and 1680 respectively. Thus A(3,3) = 1849.
%e Formatted as a triangle:
%e [1]
%e [1, 1]
%e [1, 1, 3]
%e [1, 1, 3, 9]
%e [1, 1, 7, 13, 35]
%e [1, 1, 21, 121, 75, 161]
%e [1, 1, 71, 1849, 3907, 541, 913]
%e [1, 1, 253, 35641, 426405, 202741, 4683, 6103]
%o (Sage)
%o def A260883(m, n):
%o shapes = ([x*m for x in p] for p in Partitions(n))
%o return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
%o for m in (0..4): print([A260883(m, n) for n in (0..6)])
%Y Without order: A260876.
%Y Cf. A000670, A094088, A101880, A243664, A243665, A243666, A244174.
%K nonn,tabl
%O 1,6
%A _Peter Luschny_, Aug 02 2015
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