A348454 -id:A348454 - cp's OEIS frontend

A348456 Number of ways to dissect a 2n X 2n chessboard into two polyominoes each of area 2*n^2.

1, 2, 70, 80518, 7157114189, 49852157614583644, 28289358593043414725944353, 1335056579423080371186456888543732162, 5288157175943649955880910966508435029578848399795, 1768514227824943648668138153226998430209626836775021539911012000, 50126261987194138333095266040242179892262270498222242227767710277119489194126252, 120727080026653995683405108506109122788592972611035310673809853406496349171003311517916839962975062

Offset: 0

N. J. A. Sloane, Oct 27 2021

See A348453 for much more information.
The board has 4*n^2 squares. The colors of the squares do not matter. The two parts are rook-connected polygons of area 2*n^2. They do not need to be the same polygon, only that they have the same area.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
a(4) was found on May 04 2022 by George Spahn and Manuel Kauers using an 1838 X 1838 transfer matrix found by George Spahn (see the Zeilberger link). Manuel Kauers computed the [1,2] entry of the 9th power of that matrix. The desired number a(4) is half of the coefficient of z^32 in that entry. - Doron Zeilberger, May 04 2022
Also known as the "Gerrymander Sequence" per Kauers, et al. - Michael De Vlieger, Dec 06 2022

Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
Manuel Kauers, D-Finiteness: A Success Story, Experimental Math., Johannes Kepler Univ. (Austria, 2025). See p. 6.
Manuel Kauers, Christoph Koutschan, and George Spahn, A348456(4) = 7157114189, arXiv:2209.01787 [math.CO], 2022.
Manuel Kauers, Christoph Koutschan, and George Spahn, How Does the Gerrymander Sequence Continue?, J. Int. Seq., Vol. 25 (2022), Article 22.9.7.
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).
Doron Zeilberger, Challenge to Manuel Kauers and his computer.

A column of A348452 and A348453, and a diagonal of A348454 and A348455.
See also A358289.
Cf. A167242.

Added a(5)-a(7) (from the Kauers et al. reference), Joerg Arndt, Sep 07 2022
a(8)-a(11) from Guttmann and Jensen (2022).
a(0)=1 prepended by Alois P. Heinz, Dec 06 2022

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.

Original entry on oeis.org

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

N. J. A. Sloane, Oct 27 2021

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).

			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.

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).

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.

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

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

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.

Original entry on oeis.org

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

N. J. A. Sloane, Oct 27 2021

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.

			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.

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.

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).

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

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

A348455 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 rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 27 2021

This is an essentially identical triangle to A348453, except that the data in each row has effectively been reversed. Rather than copying everything here, please refer to A348453 for further information.

			Triangle begins:
1,
1, 2, 1,
1, 10, 1,
1, 70, 117, 36, 1,
1, 4006, 1,
1, 6728, 80092, 178939, 451206, 442791, 264500, 80518, 1,
1, 158753814, 1,
1, ?, ?, 187497290034, ?, 7157114189, 1,
...

Cf. A348452, A348453, A348454, A348456.

Cf. A048691 (row lengths).

cp's OEIS Frontend

A348456 Number of ways to dissect a 2n X 2n chessboard into two polyominoes each of area 2*n^2.

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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.

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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.

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A348455 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 rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

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cp's OEIS Frontend

A348456 Number of ways to dissect a 2*n X 2*n chessboard into two polyominoes each of area 2*n^2.

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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.

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Author

Keywords

Comments

Examples

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Crossrefs

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A348455 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 rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

A348456 Number of ways to dissect a 2n X 2n chessboard into two polyominoes each of area 2*n^2.