A068393 -id:A068393 - cp's OEIS frontend

A068392 Number of partitions of n X n checkerboard into two edgewise-connected sets, counting partitions equal under rotation or reflection only once.

0, 2, 11, 92, 2100, 140834

Offset: 1

R. H. Hardin, Mar 03 2002

A265914 Number of Hamiltonian paths on an n X n grid reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

1, 1, 3, 38, 549, 28728, 1692417, 377919174, 93177169027, 91255604983167, 98333935794279062, 431583106977641773651, 2081500714709464758363648, 41476136050841717002906372881, 907951420995033325255530074961505, 82829339673122474155192677008453291270

Offset: 1

Luca Petrone, Dec 18 2015

For odd n > 1 the only symmetry possible is rotation by 180 degrees. For even n the only symmetries are reflections either horizontally or vertically. - Andrew Howroyd, Apr 15 2016

Oliver R. Bellwood, Heitor P. Casagrande, and William J. Munro, Fractal Path Strategies for Efficient 2D DMRG Simulations, arXiv:2507.11820 [cond-mat.str-el], 2025. See p. 5.
Jean-Marc Mayer, Claude Guez, and Jean Dayantis, Exact computer enumeration of the number of Hamiltonian paths in small square plane lattices, Physical Review B, Vol. 42 Number 1, 1990.

Cf. A120443, A209077, A068393.

a(9)-a(15) from Andrew Howroyd, Apr 15 2016

a(16) from Oliver R. Bellwood, Jun 06 2025

A068381 Number of partitions of n X n checkerboard by two edgewise-connected sets which produce the maximum n^2-2n+2 frontier edges between the two sets.

Original entry on oeis.org

12, 32, 96, 648, 7736, 228424, 11974112, 1599762776, 382467306272, 234367651907856, 258981528765867728, 733498025032488425464, 3770347483688546402804760, 49588653272896250824990166768

Offset: 2

R. H. Hardin, Mar 04 2002

nonn

Not divided by 4 because that property may not continue.

Each partition is counted twice in this sequence. The sequence can be computed by counting Hamiltonian paths on a n-1 x n-1 grid that start at any vertex on the grid boundary and terminate at another boundary vertex. Counts for whether the path starts or terminates on a corner or non-corner need to be computed separately as there are different multiplication factors. - Andrew Howroyd, Apr 13 2016

			Illustration of a(2)=6*2:
    __.__     __.__     __.__    __.__     __.__     __.__
   |__|  |   |  |__|   |   __|  |__   |   |__.__|   |  |  |
   |__.__|   |__.__|   |__|__|  |__|__|   |__.__|   |__|__|
Illustration of relation of a Hamiltonian path in a 3 x 3 grid to solutions of a(4):
                 .__.__.__.__.   .__.__.__.__.   .__.__.__.__.   .__.__.__.__.
   .__.__        |__.__.__.  |   |  |__.__.  |   |__.__.__.  |   |  |__.__.  |
    __.__|  <=>  |  .__.__|  |   |  .__.__|  |   |  .__.__|  |   |  .__.__|  |
   |__.__.       |  |__.__.__|   |  |__.__.__|   |  |__.__.  |   |  |__.__.  |
                 |__.__.__.__|   |__.__.__.__|   |__.__.__|__|   |__.__.__|__|

Cf. A068392, A068393.

Cf. A001184, A000532, A121789.

a(7)-a(15) from Andrew Howroyd, Apr 13 2016

cp's OEIS Frontend

A068392 Number of partitions of n X n checkerboard into two edgewise-connected sets, counting partitions equal under rotation or reflection only once.

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A265914 Number of Hamiltonian paths on an n X n grid reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A068381 Number of partitions of n X n checkerboard by two edgewise-connected sets which produce the maximum n^2-2n+2 frontier edges between the two sets.

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