cp's OEIS Frontend

A385385 Irregular triangle read by rows: T(n,k) is the number of polyominoes of size k, i.e., connected subsets of k square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= n^2.

%I A385385 #7 Jun 29 2025 11:07:40
%S A385385 1,1,1,1,1,1,1,2,3,4,4,2,1,1,1,1,2,5,10,23,44,80,87,86,49,32,10,5,1,1,
%T A385385 1,1,2,5,12,32,88,249,675,1699,3747,6993,10538,12531,11580,8458,4975,
%U A385385 2378,943,305,87,19,5,1,1
%N A385385 Irregular triangle read by rows: T(n,k) is the number of polyominoes of size k, i.e., connected subsets of k square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= n^2.
%C A385385 For n = 4, there are 384 automorphisms of the 4 X 4 torus grid graph (it is isomorphic to the 4-dimensional hypercube graph), but here we only consider the subgroup consisting of the 128 symmetries of the 4 X 4 torus. Using the full automorphism group of the torus grid graph would change row 4 to the corresponding row of A369605.
%F A385385 T(n,k) = A000105(k) if n >= k.
%F A385385 T(n,k) >= A385383(n,k)/2, with equality if and only if k = 2.
%e A385385 Triangle begins:
%e A385385   1;
%e A385385   1, 1, 1, 1;
%e A385385   1, 1, 2, 3,  4,  4,  2,  1,  1;
%e A385385   1, 1, 2, 5, 10, 23, 44, 80, 87, 86, 49, 32, 10, 5, 1, 1;
%e A385385   ...
%Y A385385 Cf. A000105, A369605, A385383 (interchange of rows and columns of the torus not allowed), A385384 (row sums), A385390 (edge subsets).
%K A385385 nonn,tabf
%O A385385 1,8
%A A385385 _Pontus von Brömssen_, Jun 27 2025