A269562 Array read by antidiagonals: T(n,m) is the number of Hamiltonian cycles in the rook graph K_n X K_m.

0, 0, 0, 1, 1, 1, 3, 3, 3, 3, 12, 30, 48, 30, 12, 60, 480, 1566, 1566, 480, 60, 360, 12000, 126120, 284112, 126120, 12000, 360, 2520, 430920, 18153720, 122330880, 122330880, 18153720, 430920, 2520, 20160, 21052080, 4357332000, 112777827840, 335750676480, 112777827840, 4357332000, 21052080, 20160

Offset

1,7

Comments

Equivalently, the number of rook tours on an n X m lattice.

2*T(n,m) is divisible by (n-1)!*(m-1)!. - Andrew Howroyd, Feb 08 2021

Links

Andrew Howroyd, Table of n, a(n) for n = 1..96

Eric Weisstein's World of Mathematics, Hamiltonian Cycle

Eric Weisstein's World of Mathematics, Rook Graph

Formula

From Andrew Howroyd, Feb 08 2021: (Start)

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

T(n,1) = (n-1)!/2 for n >= 3. (End)

Example

Array begins:
=============================================================
n\m |   1      2          3            4                5
----+--------------------------------------------------------
  1 |   0      0          1            3               12 ...
  2 |   0      1          3           30              480 ...
  3 |   1      3         48         1566           126120 ...
  4 |   3     30       1566       284112        122330880 ...
  5 |  12    480     126120    122330880     335750676480 ...
  6 |  60  12000   18153720 112777827840 2190773906150400 ...
  7 | 360 430920 4357332000 ...
     ...

Crossrefs

Column 1 is A001710(n-1) for n >= 3.

Columns 2..4 are A276356, A341498, A341499.

Main diagonal is A269561.

Cf. A269565, A286418.

Sequence in context: A349924 A170858 A024725 * A214730 A153491 A287505

Adjacent sequences: A269559 A269560 A269561 * A269563 A269564 A269565

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 29 2016

Status

approved