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A348452 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area.

%I A348452 #44 Sep 08 2022 13:09:37
%S A348452 1,1,2,0,1,1,0,10,0,0,0,0,0,1,1,70,0,117,0,0,0,36,0,0,0,0,0,0,0,1,1,0,
%T A348452 0,0,4006,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,80518,264500,
%U A348452 442791,0,451206,0,0,178939,0,0,80092,0,0,0,0,0,6728,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,158753814,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A348452 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area.
%C A348452 The board has n^2 squares. The colors do not matter. T(n,k) is zero unless k divides n^2. The tiles are rook-connected polygons made from n^2/k squares.
%C A348452 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 A348453 (the main entry for this problem) displays the same data in a more compact way (by omitting the zero entries from each row).
%C A348452 The data is taken from A004003, A172477, and Schutzman & MGGG (2018).
%H A348452 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 A348452 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 A348452 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 A348452 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 A348452 N. J. A. Sloane, <a href="/A348453/a348453_1.pdf">Illustration for T(3,3) = 10</a>
%H A348452 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 A348452 N. J. A. Sloane, <a href="/A004003/a004003_2.pdf">Illustration for T(4,8) = 36</a> [Slide from an old talk of mine]
%H A348452 N. J. A. Sloane, <a href="https://vimeo.com/704569041/4ffa06b95e">The On-Line Encyclopedia of Integer Sequences: An illustrated guide with many unsolved problems</a>, Guest Lecture given in Doron Zeilberger's Experimental Mathematics Math640 Class, Rutgers University, Spring Semester, Apr 28 2022: <a href="https://sites.math.rutgers.edu/~zeilberg/EM22/C27.pdf">Slides</a>; <a href="http://NeilSloane.com/doc/Math640.04.2022.pdf">Slides (an alternative source)</a>.
%F A348452 A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.
%e A348452 The first seven rows of the triangle are:
%e A348452 1,
%e A348452 1, 2, 0, 1,
%e A348452 1, 0, 10, 0, 0, 0, 0, 0, 1,
%e A348452 1, 70, 0, 117, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 1,
%e A348452 1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A348452 1, 80518, 264500, 442791, 0, 451206, 0, 0, 178939, 0, 0, 80092, 0, 0, 0, 0, 0, 6728, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A348452 1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A348452 ...
%e A348452 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 A348452 Cf. A348453. A348454 and A348455 are similar triangles with the data in each row reversed. The row sums are in A348789.
%Y A348452 See also A004003, A099390, A172477, A348456.
%K A348452 nonn,tabf
%O A348452 1,3
%A A348452 _N. J. A. Sloane_, Oct 27 2021
%E A348452 More than the usual number of terms are given, in order to show the first seven rows.