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%I #51 Jan 14 2023 09:55:32
%S 1,1,2,1,1,10,1,1,70,117,36,1,1,4006,1,1,80518,264500,442791,451206,
%T 178939,80092,6728,1,1,158753814,1,7157114189
%N Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.
%C The board has n^2 squares. The colors do not matter. The tiles are rook-connected polygons made from n^2/d_k squares.
%C This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
%C A348452 displays the same data in a less compact way. The present triangle is obtained by omitting the zero entries from A348452.
%C The data is taken from A004003, A172477, A348456, and Schutzman & MGGG (2018).
%C T(8,2) = 7157114189 (see A348456). T(8,3) is presently unknown.
%H Moon Duchin, <a href="https://www.youtube.com/watch?v=VU8CtVmiP3w">Graphs, Geometry and Gerrymandering</a>”, Talk at Celebration of Mind Conference, Oct 23 2021.
%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1016/0031-8914(61)90063-5">The Statistics of Dimers on a Lattice</a>, Physica, 27 (1961), 1209-1225.
%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1063/1.1703953">Dimer statistics and phase transitions</a>, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
%H Zach Schutzman and MGGG, <a href="https://mggg.org/table.html">The Known Sizes of Grid Metagraphs</a>, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.
%H N. J. A. Sloane, <a href="/A348453/a348453_1.pdf">Illustration for T(3,2) = 10</a>
%H N. J. A. Sloane, <a href="/A348453/a348453.pdf">Illustration for T(4,2) = 70</a> [Labels give code, B = length of internal boundary, C = number of internal corners, G = group order, # = number of this type. Note that (B,C) determines the type]
%H N. J. A. Sloane, <a href="/A004003/a004003_2.pdf">Illustration for T(4,4) = 36</a> [Slide from an old talk of mine]
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 21.
%F A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.
%e The first eight rows of the triangle are:
%e 1,
%e 1, 2, 1,
%e 1, 10, 1,
%e 1, 70, 117, 36, 1,
%e 1, 4006, 1,
%e 1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1,
%e 1, 158753814, 1,
%e 1, 7157114189, ?, 187497290034, ?, ?, 1,
%e ...
%e The corresponding divisors d_k are:
%e 1,
%e 1, 2, 4,
%e 1, 3, 9,
%e 1, 2, 4, 8, 16,
%e 1, 5, 25,
%e ...
%e The domino is the only polyomino of area 2, and the 36 ways to tile a 4 X 4 square with dominoes are shown in one of the links.
%Y Cf. A348452. A348454 and A348455 are similar triangles with the data in each row reversed.
%Y See also A004003, A099390, A172477, A348456.
%Y Cf. A048691 (row lengths).
%K nonn,tabf,more
%O 1,3
%A _N. J. A. Sloane_, Oct 27 2021.
%E T(8,2) added May 04 2022 (see A348456) - _N. J. A. Sloane_, May 05 2022
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