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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.
3

%I #22 Mar 17 2018 04:13:15

%S 2,3,7,44,494,748827,99987552,23904291912,23904291912,14647978829979,

%T 16186345621426754,45843626565163628751,235646717730827228414584,

%U 3099290829556018890177304005

%N 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.

%C 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

%e From _Andrew Howroyd_, Apr 15 2016: (Start)

%e Case n=4: There are 2 nonisomorphic symmetrical solutions (see illustration below). a(4)=(A068381(4)/8 + 2)/2 = 7.

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%e Case n=5: There are 7 nonisomorphic symmetrical solutions (see illustration below). a(5)=(A068381(5)/8 + 7)/2 = 44.

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%e (End)

%Y Cf. A068381, A068416, A068392, A265914.

%K nonn

%O 2,1

%A _R. H. Hardin_, Mar 03 2002

%E a(7)-a(15) from _Andrew Howroyd_, Apr 15 2016