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.

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, 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, 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

Offset

1,3

Comments

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.

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

A348453 (the main entry for this problem) displays the same data in a more compact way (by omitting the zero entries from each row).

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

Links

Table of n, a(n) for n=1..140.

Moon Duchin, Graphs, Geometry and Gerrymandering”, Talk at Celebration of Mind Conference, Oct 23 2021.

P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.

P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).

Zach Schutzman and MGGG, The Known Sizes of Grid Metagraphs, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.

N. J. A. Sloane, Illustration for T(3,3) = 10

N. J. A. Sloane, Illustration for T(4,2) = 70 [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]

N. J. A. Sloane, Illustration for T(4,8) = 36 [Slide from an old talk of mine]

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences: An illustrated guide with many unsolved problems, Guest Lecture given in Doron Zeilberger's Experimental Mathematics Math640 Class, Rutgers University, Spring Semester, Apr 28 2022: Slides; Slides (an alternative source).

Formula

A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.

Example

The first seven rows of the triangle are:
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, 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, 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,
...
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.

Crossrefs

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

See also A004003, A099390, A172477, A348456.

Sequence in context: A269953 A326411 A348454 * A309163 A320655 A359786

Adjacent sequences: A348449 A348450 A348451 * A348453 A348454 A348455

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 27 2021

Extensions

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

Status

approved