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%I #20 Sep 02 2022 01:28:40
%S 1,1,1,6,1,30,90,1,56,70,1260,2520,1,90,420,3780,9450,75600,113400,1,
%T 132,990,924,8910,83160,34650,332640,1247400,6237000,7484400,1,182,
%U 2002,6006,18018,270270,252252,630630,1081080,15135120,12612600,37837800,189189000,681080400,681080400
%N Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.
%C We call an irregular triangle T a partition triangle if T(n, k) is defined for n >= 0 and 0 <= k < A000041(n).
%C T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 2. For instance 2*P(4, .) = [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]].
%e Triangle starts (note the subdivisions by ';' (A072233)):
%e [0] [1]
%e [1] [1]
%e [2] [1; 6]
%e [3] [1; 30; 90]
%e [4] [1; 56, 70; 1260; 2520]
%e [5] [1; 90, 420; 3780, 9450; 75600; 113400]
%e [6] [1; 132, 990, 924; 8910, 83160, 34650; 332640, 1247400; 6237000; 7484400]
%e .
%e T(4, 1) = 56 because [6, 2] is the integer partition 2*P(4, 1) in the canonical order and there are 28 set partitions which have the shape [6, 2] (an example is {{1, 3, 4, 5, 6, 8}, {2, 7}}). Finally, since the order of the sets is taken into account, one gets 2!*28 = 56.
%o (Sage)
%o def GenOrdSetPart(m, n):
%o shapes = ([x*m for x in p] for p in Partitions(n))
%o return [factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes]
%o def A327022row(n): return GenOrdSetPart(2, n)
%o for n in (0..6): print(A327022row(n))
%Y Row sums: A094088, alternating row sums: A028296, main diagonal: A000680, central column A281478, by length: A241171.
%Y Cf. A178803 (m=0), A133314 (m=1), this sequence (m=2), A327023 (m=3), A327024 (m=4).
%Y Cf. A080577, A000041, A072233.
%K nonn,tabf
%O 0,4
%A _Peter Luschny_, Aug 27 2019
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