A358289 -id:A358289 - 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

A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

Original entry on oeis.org

0, 6, 53, 627, 16213, 1123743, 221984391, 127561384993, 215767063451331, 1082828220389781579, 16209089366362071416785, 726438398002211876667379681, 97741115155002465272674416929195, 39565596445488219947994403962984729307

Offset: 1

R. H. Hardin, Mar 02 2002

nonn

One of the partitions may completely surround the other. (Cf. A271802) - Andrew Howroyd, Apr 14 2016

Number of minimal edge cuts in the n X n grid graph. - Andrew Howroyd, Dec 11 2024

			Illustration of a(2)=6:
   11   12   12   12   11   11
   22   12   22   11   12   21
Illustration of a few solutions of a(3):
   111   112   122   111   111
   121   111   112   212   111
   111   111   222   222   222

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 1..26
Benjamin Fifield, Kosuke Imai, Jun Kawahara, and Christopher T. Kenny, The Essential Role of Empirical Validation in Legislative Redistricting Simulation, Tech. rep., Department of Government and Department of Statistics, Harvard University (2019).
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cut.

Cf. A068392, A166755, A271802, A068381, A068417, A113900, A348456, A358289.

a(n) = A271802(n) + A140517(n-2). - Andrew Howroyd, Apr 14 2016

a(n) = A166755(n)/2. - Andrew Howroyd, Dec 11 2024

a(7)-a(14) from Andrew Howroyd, Apr 14 2016

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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A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

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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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A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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