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.
T(8,2) = 7157114189 (see A348456). T(8,3) is presently unknown.
LINKS
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
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. A048691 (row lengths).
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