A395327 - OEIS

A395327

Triangle read by rows: T(n,k) is the number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles with k orbits of blocks.

1, 0, 2, 0, 4, 4, 0, 10, 24, 8, 0, 28, 128, 96, 16, 0, 82, 680, 880, 320, 32, 0, 244, 3664, 7440, 4480, 960, 64, 0, 730, 20104, 61208, 54880, 19040, 2688, 128, 0, 2188, 112368, 500416, 632576, 318080, 71680, 7168, 256, 0, 6562, 638760, 4104640, 7082880, 4869312, 1564416, 247296, 18432, 512

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OFFSET

0,3

LINKS

Alois P. Heinz, Rows n = 0..150, flattened

FORMULA

E.g.f.: exp( y*((exp(3*x) - 1)/3 + exp(x) - 1 ) ).

EXAMPLE

Triangle begins:

0, 2;

0, 4, 4;

0, 10, 24, 8;

0, 28, 128, 96, 16;

0, 82, 680, 880, 320, 32;

0, 244, 3664, 7440, 4480, 960, 64;

0, 730, 20104, 61208, 54880, 19040, 2688, 128;

0, 2188, 112368, 500416, 632576, 318080, 71680, 7168, 256;

...

Case n=2: Without loss of generality the permutation of two 3-cycles can be taken as (123)(456). The T(2,1) = 4 set partitions that are invariant under this permutation with 1 orbit are {{1, 2, 3, 4, 5, 6}}, {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 6}, {2, 4}, {3, 5}}. The T(2,2) = 4 partitions with 2 orbits are {{1, 2, 3}, {4, 5, 6}}, {{1, 2, 3}, {4}, {5}, {6}}, {{1}, {2}, {3}, {4, 5, 6}}, {{1}, {2}, {3}, {4}, {5}, {6}}.

MAPLE

b:= proc(n) option remember; `if`(n=0, 1, add((1+

3^(j-1))*binomial(n-1, j-1)*expand(b(n-j)*x), j=1..n))

end:

T:= (n, k)-> coeff(b(n), x, k):

seq(seq(T(n, k), k=0..n), n=0..9); # Alois P. Heinz, Apr 27 2026

PROG

(PARI) T(n) = {[Vecrev(p) | p<-Vec(serlaplace(exp( y*((exp(3*x + O(x*x^n)) - 1)/3 + exp(x + O(x*x^n)) - 1 ) )))]}

{ my(A=T(8)); for(i=1, #A, print(A[i])) }

CROSSREFS

Row sums are A002874.

Maximum row values are A002875.