%I #47 Apr 21 2024 23:54:29
%S 1,2,2,6,8,6,24,60,60,24,120,816,1512,816,120,720,17520,83520,83520,
%T 17520,720,5040,550080,8869680,22394880,8869680,550080,5040,40320,
%U 23839200,1621680480,13346910720,13346910720,1621680480,23839200,40320
%N Array read by antidiagonals: T(n,m) is the number of (directed) Hamiltonian paths in K_n X K_m.
%C Equivalently, the number of directed Hamiltonian paths on the n X m rook graph.
%C Conjecture: T(n,m) mod n!*m! = 0. - _Mikhail Kurkov_, Feb 08 2019
%C The above conjecture is true since a path defines an ordering on the rows and columns by the order in which they are first visited by the path. Every permutation of rows and columns therefore gives a different path. - _Andrew Howroyd_, Feb 08 2021
%H Andrew Howroyd, <a href="/A269565/b269565.txt">Table of n, a(n) for n = 1..96</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%F From _Andrew Howroyd_, Oct 20 2019: (Start)
%F T(n,m) = T(m,n).
%F T(n,1) = n!. (End)
%e Array begins:
%e ===========================================================
%e n\m| 1 2 3 4 5
%e ---+-------------------------------------------------------
%e 1 | 1, 2, 6, 24, 120, ...
%e 2 | 2, 8, 60, 816, 17520, ...
%e 3 | 6, 60, 1512, 83520, 8869680, ...
%e 4 | 24, 816, 83520, 22394880, 13346910720, ...
%e 5 | 120, 17520, 8869680, 13346910720, 50657369241600, ...
%e ...
%Y Main diagonal is A096970.
%Y Columns 2..3 are A096121, A329319.
%Y Cf. A286418, A269562.
%K nonn,tabl
%O 1,2
%A _Andrew Howroyd_, Feb 29 2016