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]].
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Aug 27 2019
STATUS
approved