A269565 Array read by antidiagonals: T(n,m) is the number of (directed) Hamiltonian paths in K_n X K_m.

1, 2, 2, 6, 8, 6, 24, 60, 60, 24, 120, 816, 1512, 816, 120, 720, 17520, 83520, 83520, 17520, 720, 5040, 550080, 8869680, 22394880, 8869680, 550080, 5040, 40320, 23839200, 1621680480, 13346910720, 13346910720, 1621680480, 23839200, 40320

Offset: 1

Author

Andrew Howroyd, Feb 29 2016

Keywords

• nonn
• tabl

Comments

Equivalently, the number of directed Hamiltonian paths on the n X m rook graph.

Conjecture: T(n,m) mod n!*m! = 0. - Mikhail Kurkov, Feb 08 2019

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

Examples

			Array begins:
===========================================================
n\m|    1      2        3            4               5
---+-------------------------------------------------------
1  |    1,     2,       6,          24,            120, ...
2  |    2,     8,      60,         816,          17520, ...
3  |    6,    60,    1512,       83520,        8869680, ...
4  |   24,   816,   83520,    22394880,    13346910720, ...
5  |  120, 17520, 8869680, 13346910720, 50657369241600, ...
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..96
• Eric Weisstein's World of Mathematics, Hamiltonian Path
• Eric Weisstein's World of Mathematics, Rook Graph

Crossrefs

Main diagonal is A096970.

Columns 2..3 are A096121, A329319.

Cf. A286418, A269562.

Formula

From Andrew Howroyd, Oct 20 2019: (Start)

T(n,m) = T(m,n).

T(n,1) = n!. (End)