A068416 -id:A068416 - cp's OEIS frontend

A378932 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the grid graph P_m X P_n.

0, 1, 1, 2, 6, 2, 3, 15, 15, 3, 4, 28, 53, 28, 4, 5, 45, 146, 146, 45, 5, 6, 66, 356, 627, 356, 66, 6, 7, 91, 809, 2471, 2471, 809, 91, 7, 8, 120, 1759, 9292, 16213, 9292, 1759, 120, 8, 9, 153, 3716, 33878, 103196, 103196, 33878, 3716, 153, 9, 10, 190, 7702, 120771, 642364, 1123743, 642364, 120771, 7702, 190, 10

Offset: 1

Andrew Howroyd, Dec 11 2024

T(m,n) is the number of partitionings of an m X n checkerboard into two edgewise-connected sets.

			Table starts:
===================================================
m\n | 1  2    3     4      5        6         7 ...
----+----------------------------------------------
  1 | 0  1    2     3      4        5         6 ...
  2 | 1  6   15    28     45       66        91 ...
  3 | 2 15   53   146    356      809      1759 ...
  4 | 3 28  146   627   2471     9292     33878 ...
  5 | 4 45  356  2471  16213   103196    642364 ...
  6 | 5 66  809  9292 103196  1123743  12028981 ...
  7 | 6 91 1759 33878 642364 12028981 221984391 ...
  ...

Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cut.

Main diagonal is A068416.
Rows 1..4 are A001477(n-1), A000384, A378933, A378934.
Rows 3..8 multiplied by 2 are A166761, A166766, A166769, A166771, A166773, A166774.
Cf. A287151, A359990.

T(m,n) = T(n,m).

A068393 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. Partitions equal under rotation or reflection are counted only once.

Original entry on oeis.org

2, 3, 7, 44, 494, 748827, 99987552, 23904291912, 23904291912, 14647978829979, 16186345621426754, 45843626565163628751, 235646717730827228414584, 3099290829556018890177304005

Offset: 2

R. H. Hardin, Mar 03 2002

nonn

For even n > 2 the only symmetry possible is rotation by 180 degrees. For odd n > 1 the only symmetries are reflections either horizontally or vertically. - Andrew Howroyd, Apr 15 2016

			From _Andrew Howroyd_, Apr 15 2016: (Start)
Case n=4: There are 2 nonisomorphic symmetrical solutions (see illustration below). a(4)=(A068381(4)/8 + 2)/2 = 7.
    __.__.__.__.    __.__.__.__.
   |   __    __|   |   __   |  |
   |  |  |  |  |   |  |  |  |  |
   |__|  |__|  |   |  |  |__|  |
   |__.__.__.__|   |__|__.__.__|
Case n=5: There are 7 nonisomorphic symmetrical solutions (see illustration below). a(5)=(A068381(5)/8 + 7)/2 = 44.
    __.__.__.__.__.   __.__.__.__.__.   __.__.__.__.__.   __.__.__.__.__.
   |   __|  |__   |  |   __|  |__   |  |  |__    __|  |  |  |   __   |  |
   |  |__    __|  |  |  |   __   |  |  |   __|  |__   |  |  |  |  |  |  |
   |   __|  |__   |  |  |  |  |  |  |  |  |   __   |  |  |  |  |  |  |  |
   |  |__.__.__|  |  |  |__|  |__|  |  |  |__|  |__|  |  |  |__|  |__|  |
   |__.__.__.__.__|  |__.__.__.__.__|  |__.__.__.__.__|  |__.__.__.__.__|
    __.__.__.__.__.   __.__.__.__.__.   __.__.__.__.__.
   |__.__    __.__|  |__    __    __|  |   __    __   |
   |   __|  |__   |  |  |  |  |  |  |  |__|  |  |  |__|
   |  |   __   |  |  |  |  |  |  |  |  |   __|  |__   |
   |  |__|  |__|  |  |  |__|  |__|  |  |  |__.__.__|  |
   |__.__.__.__.__|  |__.__.__.__.__|  |__.__.__.__.__|
(End)

Cf. A068381, A068416, A068392, A265914.

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

A166755 Number of n X n 1..2 arrays containing at least one of each value, and all equal values connected.

Original entry on oeis.org

0, 12, 106, 1254, 32426, 2247486, 443968782, 255122769986, 431534126902662, 2165656440779563158, 32418178732724142833570, 1452876796004423753334759362, 195482230310004930545348833858390, 79131192890976439895988807925969458614, 96533107106359142781127117074385161767893162

Offset: 1

R. H. Hardin, Oct 21 2009

nonn

			Some solutions for n=4
...1.2.2.2...1.1.1.1...2.2.2.1...2.1.1.1...1.1.1.1...2.2.2.2...1.1.1.2
...1.2.1.2...2.2.2.1...2.2.2.1...2.2.1.1...1.1.2.2...1.2.2.2...2.2.1.2
...1.1.1.2...2.2.2.1...2.2.1.1...1.1.1.1...1.1.2.2...2.2.2.2...2.1.1.2
...2.2.2.2...2.1.1.1...2.2.2.2...1.1.1.1...2.2.2.2...2.2.2.2...2.2.2.2
------
...1.1.1.2...2.1.1.1...2.2.2.2...1.1.1.1...1.1.1.1...1.1.1.1...2.2.2.2
...1.1.2.2...2.1.1.1...2.2.1.2...1.1.1.1...2.2.1.1...1.2.1.2...1.2.2.2
...1.1.1.2...2.2.1.1...2.2.1.2...1.1.2.1...1.2.1.1...1.2.2.2...1.1.2.2
...1.1.2.2...2.2.2.2...2.1.1.1...2.2.2.1...1.1.1.1...1.1.1.2...1.1.1.2

Andrew Howroyd, Table of n, a(n) for n = 1..26

Cf. A068416.

a(n) = 2*A068416(n). - Andrew Howroyd, Dec 11 2024

a(7) onwards added using A068416 by Andrew Howroyd, Dec 11 2024

A271802 Number of cuttings of an n X n checkerboard along grid lines into two pieces with holes disallowed.

Original entry on oeis.org

0, 6, 52, 614, 16000, 1114394, 220762028, 127074234622, 215163221802400, 1080509693050320314, 16181730102294154610684, 725449589191165593072311582, 97631783799192329642727718567824, 39528641527526180063041016094650084850

Offset: 1

Andrew Howroyd, Apr 14 2016

nonn

Equivalently, the number of partitionings of an n X n checkerboard into two edgewise-connected simply-connected sets. (Cf. A068416).

Each part is required to contain at least one cell and cuttings are considered different if they only differ by rotation or reflection.

A068416 = Cases[Import["https://oeis.org/A068416/b068416.txt", "Table"], {, }][[All, 2]];
A140517 = Cases[Import["https://oeis.org/A140517/b140517.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 1, 0, A068416[[n]] - A140517[[n - 1]]];
Array[a, 14] (* Jean-François Alcover, Sep 15 2019 *)

a(n) = A068416(n) - A140517(n-2).

cp's OEIS Frontend

A378932 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the grid graph P_m X P_n.

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A068393 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. Partitions equal under rotation or reflection are counted only once.

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A166755 Number of n X n 1..2 arrays containing at least one of each value, and all equal values connected.

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Author

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Examples

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Crossrefs

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Extensions

A271802 Number of cuttings of an n X n checkerboard along grid lines into two pieces with holes disallowed.

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