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A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

%I A327023 #23 Mar 23 2020 02:43:05
%S A327023 1,1,1,20,1,168,1680,1,440,924,55440,369600,1,910,10010,300300,
%T A327023 1261260,33633600,168168000,1,1632,37128,48620,1113840,24504480,
%U A327023 17153136,326726400,2058376320,34306272000,137225088000
%N A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.
%C A327023 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 = 3. For instance 3*P(4, .) = [[12], [9, 3], [6, 6], [6, 3, 3], [3, 3, 3, 3]].
%e A327023 Triangle starts (note the subdivisions by ';' (A072233)):
%e A327023 [0] [1]
%e A327023 [1] [1]
%e A327023 [2] [1;   20]
%e A327023 [3] [1;  168;  1680]
%e A327023 [4] [1;  440,   924;  55440;  369600]
%e A327023 [5] [1;  910, 10010; 300300, 1261260; 33633600; 168168000]
%e A327023 [6] [1; 1632, 37128,  48620; 1113840, 24504480,  17153136; 326726400, 2058376320;
%e A327023      34306272000; 137225088000]
%e A327023 .
%e A327023 T(4, 1) = 440 because [9, 3] is the integer partition 3*P(4, 1) in the canonical order and there are 220 set partitions which have the shape [9, 3]. Finally, since the order of the sets is taken into account, one gets 2!*220 = 440.
%o A327023 (Sage) # uses[GenOrdSetPart from A327022]
%o A327023 def A327023row(n): return GenOrdSetPart(3, n)
%o A327023 for n in (0..6): print(A327023row(n))
%Y A327023 Row sums: A243664, alternating row sums: A002115, main diagonal: A014606, central column A281479, by length: A278073.
%Y A327023 Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), this sequence (m=3), A327024 (m=4).
%Y A327023 Cf. A080577, A000041, A072233.
%K A327023 nonn,tabf
%O A327023 0,4
%A A327023 _Peter Luschny_, Aug 27 2019