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Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n with respect to the symmetries of the square.
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%I #26 Jul 22 2024 09:47:36

%S 1,1,1,1,1,1,1,1,2,5,5,5,2,1,1,7,56,317,1524,5733,17728,44767,94427,

%T 166786,249624,316950,343424,316950,249624,166786,94427,44767,17728,

%U 5733,1524,317,56,7,1,1,23,1012,36125,1035496,23878229,456936220,7437730463

%N Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n with respect to the symmetries of the square.

%C A permutation, p, can be thought of as a set of points (i, p(i)). In this viewpoint it is natural to consider the symmetries of the square.

%C T(n,k) is the number of symmetry classes of subsets of size k from S_n.

%H Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, <a href="https://doi.org/10.37236/12686">Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding</a>, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); <a href="https://arxiv.org/abs/2312.07716">arXiv preprint</a>, arXiv:2312.07716 [math.CO], 2023.

%F T(n,k) = 1/8 * (C(n,k) + 2*A277080(n,k) + 2*A277081(n,k) + 2*A277085(n,k) + A277083(n,k)).

%e Triangle starts:

%e 1, 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 5, 5, 5, 2, 1;

%Y Rows lengths give A038507.

%Y Cf. A277080, A277081, A277083, A277085.

%K nonn,tabf

%O 0,9

%A _Christian Bean_, Sep 28 2016