A348453 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.

1, 1, 2, 1, 1, 10, 1, 1, 70, 117, 36, 1, 1, 4006, 1, 1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1, 1, 158753814, 1, 7157114189

Offset

1,3

Comments

The board has n^2 squares. The colors do not matter. The tiles are rook-connected polygons made from n^2/d_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.

A348452 displays the same data in a less compact way. The present triangle is obtained by omitting the zero entries from A348452.

The data is taken from A004003, A172477, A348456, and Schutzman & MGGG (2018).

T(8,2) = 7157114189 (see A348456). T(8,3) is presently unknown.

Links

Table of n, a(n) for n=1..28.

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(3,2) = 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(4,4) = 36 [Slide from an old talk of mine]

N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.

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 eight rows of the triangle are:
  1,
  1, 2, 1,
  1, 10, 1,
  1, 70, 117, 36, 1,
  1, 4006, 1,
  1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1,
  1, 158753814, 1,
  1, 7157114189, ?, 187497290034, ?, ?, 1,
  ...
The corresponding divisors d_k are:
  1,
  1, 2, 4,
  1, 3, 9,
  1, 2, 4, 8, 16,
  1, 5, 25,
  ...
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. A348452. A348454 and A348455 are similar triangles with the data in each row reversed.

See also A004003, A099390, A172477, A348456.

Cf. A048691 (row lengths).

Sequence in context: A104251 A320576 A348455 * A345748 A153731 A262226

Adjacent sequences: A348450 A348451 A348452 * A348454 A348455 A348456

Keyword

nonn,tabf,more

Author

N. J. A. Sloane, Oct 27 2021.

Extensions

T(8,2) added May 04 2022 (see A348456) - N. J. A. Sloane, May 05 2022

Status

approved