A348452 - OEIS

%I #44 Sep 08 2022 13:09:37

%S 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 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 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 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 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 This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.

%C 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 The data is taken from A004003, A172477, and Schutzman & MGGG (2018).

%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(3,3) = 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(4,8) = 36</a> [Slide from an old talk of mine]

%H 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 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 seven rows of the triangle are:

%e 1,

%e 1, 2, 0, 1,

%e 1, 0, 10, 0, 0, 0, 0, 0, 1,

%e 1, 70, 0, 117, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 1,

%e 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 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 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 ...

%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. A348453. A348454 and A348455 are similar triangles with the data in each row reversed. The row sums are in A348789.

%Y See also A004003, A099390, A172477, A348456.

%K nonn,tabf

%O 1,3

%A _N. J. A. Sloane_, Oct 27 2021

%E More than the usual number of terms are given, in order to show the first seven rows.