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A270227 Array read by antidiagonals: T(n,m) is the number of matchings in the rook graph K_n X K_m.

%I A270227 #21 Feb 16 2025 08:33:31
%S A270227 1,2,2,4,7,4,10,32,32,10,26,193,370,193,26,76,1382,5950,5950,1382,76,
%T A270227 232,11719,122984,270529,122984,11719,232,764,112604,3175696,16873930,
%U A270227 16873930,3175696,112604,764,2620,1221889,98815588,1384880065,3337807996,1384880065,98815588,1221889,2620
%N A270227 Array read by antidiagonals: T(n,m) is the number of matchings in the rook graph K_n X K_m.
%C A270227 Observations: (for n+m <= 32)
%C A270227 Examination of values modulus a small prime yields several patterns.
%C A270227 T(n,m) == (n+1)*(m+1) (mod 2) for n+m>2.
%C A270227 T(n,m) == T(n,m+6) (mod 3).
%C A270227 T(n,m) is not divisible by 3.
%C A270227 T(n,m) == 0 (mod 5) for n==4 (mod 5) and m<>2 and except when m=n=4.
%C A270227 T(5,m) == 0 (mod 208) for m >= 13.
%C A270227 T(6,m) == 0 (mod 19) for m >= 19.
%H A270227 Andrew Howroyd, <a href="/A270227/b270227.txt">Table of n, a(n) for n = 1..496</a>
%H A270227 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H A270227 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching.html">Matching</a>
%H A270227 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%e A270227 The start of the sequence as table:
%e A270227 *   1     2       4         10           26              76 ...
%e A270227 *   2     7      32        193         1382           11719 ...
%e A270227 *   4    32     370       5950       122984         3175696 ...
%e A270227 *  10   193    5950     270529     16873930      1384880065 ...
%e A270227 *  26  1382  122984  168739305   3337807996    909046586596 ...
%e A270227 *  76 11719 3175696 1384880065 909046586596 855404716021831 ...
%e A270227 * ...
%Y A270227 Main diagonal is A270228. Rows include A000085, A270229.
%K A270227 nonn,tabl
%O A270227 1,2
%A A270227 _Andrew Howroyd_, Mar 13 2016