cp's OEIS Frontend

A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

%I A327024 #20 Mar 23 2020 02:43:11
%S A327024 1,1,1,70,1,990,34650,1,3640,12870,2702700,63063000,1,9690,251940,
%T A327024 26453700,187065450,17459442000,305540235000,1,21252,1470942,2704156,
%U A327024 154448910,8031343320,9465511770,374796021600,3975514943400,231905038365000,3246670537110000
%N A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.
%C A327024 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 = 4. For instance 4*P(4, .) = [[16], [12, 4], [8, 8], [8, 4, 4], [4, 4, 4, 4]].
%e A327024 Triangle starts (note the subdivisions by ';' (A072233)):
%e A327024 [0] [1]
%e A327024 [1] [1]
%e A327024 [2] [1;    70]
%e A327024 [3] [1;   990;   34650]
%e A327024 [4] [1;  3640,   12870;  2702700;  63063000]
%e A327024 [5] [1;  9690,  251940; 26453700, 187065450; 17459442000; 305540235000]
%e A327024 [6] [1; 21252, 1470942,  2704156; 154448910,  8031343320,   9465511770;
%e A327024      374796021600, 3975514943400; 231905038365000; 3246670537110000]
%e A327024 .
%e A327024 T(4, 1) = 3640 because [12, 4] is the integer partition 4*P(4, 1) in the canonical order and there are 1820 set partitions which have the shape [12, 4]. Finally, since the order of the sets is taken into account, one gets 2!*1820 = 3640.
%o A327024 (Sage) # uses[GenOrdSetPart from A327022]
%o A327024 def A327024row(n): return GenOrdSetPart(4, n)
%o A327024 for n in (0..6): print(A327024row(n))
%Y A327024 Row sums: A243665, alternating row sums: A211212, main diagonal: A014608, central column: A281480, by length: A278074.
%Y A327024 Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), A327023 (m=3), this sequence (m=4).
%Y A327024 Cf. A080577, A000041, A072233.
%K A327024 nonn,tabf
%O A327024 0,4
%A A327024 _Peter Luschny_, Aug 27 2019