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Array read by antidiagonals: T(n,m) is the number of (directed) Hamiltonian paths in K_n X K_m.
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%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