A395327 - OEIS

%I #12 Apr 27 2026 10:23:40

%S 1,0,2,0,4,4,0,10,24,8,0,28,128,96,16,0,82,680,880,320,32,0,244,3664,

%T 7440,4480,960,64,0,730,20104,61208,54880,19040,2688,128,0,2188,

%U 112368,500416,632576,318080,71680,7168,256,0,6562,638760,4104640,7082880,4869312,1564416,247296,18432,512

%N 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.

%H Alois P. Heinz, <a href="/A395327/b395327.txt">Rows n = 0..150, flattened</a>

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

%e Triangle begins:

%e   1;

%e   0,    2;

%e   0,    4,      4;

%e   0,   10,     24,      8;

%e   0,   28,    128,     96,     16;

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

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

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

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

%e   ...

%e 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}}.

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

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

%p     end:

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

%p seq(seq(T(n,k), k=0..n), n=0..9);  # _Alois P. Heinz_, Apr 27 2026

%o (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 ) )))]}

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

%Y Row sums are A002874.

%Y Maximum row values are A002875.

%Y Main diagonal is A000079.

%Y Column k=1 is A034472(n-1).

%Y Cf. A294201, A395326.

%K nonn,tabl

%O 0,3

%A _Andrew Howroyd_ and _Andrei Zabolotskii_, Apr 19 2026