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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. 3
 1, 1, 1, 20, 1, 168, 1680, 1, 440, 924, 55440, 369600, 1, 910, 10010, 300300, 1261260, 33633600, 168168000, 1, 1632, 37128, 48620, 1113840, 24504480, 17153136, 326726400, 2058376320, 34306272000, 137225088000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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]]. LINKS Table of n, a(n) for n=0..29. EXAMPLE Triangle starts (note the subdivisions by ';' (A072233)): [0] [1] [1] [1] [2] [1; 20] [3] [1; 168; 1680] [4] [1; 440, 924; 55440; 369600] [5] [1; 910, 10010; 300300, 1261260; 33633600; 168168000] [6] [1; 1632, 37128, 48620; 1113840, 24504480, 17153136; 326726400, 2058376320; 34306272000; 137225088000] . 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. PROG (Sage) # uses[GenOrdSetPart from A327022] def A327023row(n): return GenOrdSetPart(3, n) for n in (0..6): print(A327023row(n)) CROSSREFS Row sums: A243664, alternating row sums: A002115, main diagonal: A014606, central column A281479, by length: A278073. Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), this sequence (m=3), A327024 (m=4). Cf. A080577, A000041, A072233. Sequence in context: A200092 A164812 A332258 * A280621 A223522 A040395 Adjacent sequences: A327020 A327021 A327022 * A327024 A327025 A327026 KEYWORD nonn,tabf AUTHOR Peter Luschny, Aug 27 2019 STATUS approved

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Last modified August 13 01:30 EDT 2024. Contains 375113 sequences. (Running on oeis4.)