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Number of (non-null) connected induced subgraphs in the n X n rook complement graph.
2

%I #16 Sep 19 2017 05:39:49

%S 1,6,397,64627,33548446,68719441230,562949953224709,

%T 18446744073708514623,2417851639229258344134994,

%U 1267650600228229401496677070990,2658455991569831745807614120434011325,22300745198530623141535718272648360902487971

%N Number of (non-null) connected induced subgraphs in the n X n rook complement graph.

%C From _Andrew Howroyd_, Aug 30 2017: (Start)

%C The vertex sets inducing disconnected subgraphs are:

%C - two or more vertices taken from a single row or column,

%C - any vertex combined with at least one from the same row and at least one from the same column,

%C - four vertices forming the corners of a rectangle. (End)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>

%F a(n) = 2^(n^2) - 2*n*(2^n-n-1) - n^2*(2^(n-1)-1)^2 - binomial(n,2)^2 - 1. - _Andrew Howroyd_, Aug 30 2017

%t Table[2^(n^2) - 2 n (2^n - n - 1) - n^2 (2^(n - 1) - 1)^2 - Binomial[n, 2]^2 - 1, {n, 10}]

%o (PARI) a(n) = 2^(n^2) - 2*n*(2^n-n-1) - n^2*(2^(n-1)-1)^2 - binomial(n,2)^2 - 1; \\ _Andrew Howroyd_, Aug 30 2017

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Aug 27 2017

%E a(6)-a(12) from _Andrew Howroyd_, Aug 30 2017