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Eulerian Cycles

Problem 289

Published on Friday, 23rd April 2010, 01:00 pm; Solved by 360

Let C(x,y) be a circle passing through the points (x, y), (x, y+1), (x+1, y) and (x+1, y+1).

For positive integers m and n, let E(m,n) be a configuration which consists of the m·n circles:
{ C(x,y): 0 ≤ x < m, 0 ≤ y < n, x and y are integers }

An Eulerian cycle on E(m,n) is a closed path that passes through each arc exactly once.
Many such paths are possible on E(m,n), but we are only interested in those which are not self-crossing: A non-crossing path just touches itself at lattice points, but it never crosses itself.

The image below shows E(3,3) and an example of an Eulerian non-crossing path.

Let L(m,n) be the number of Eulerian non-crossing paths on E(m,n).
For example, L(1,2) = 2, L(2,2) = 37 and L(3,3) = 104290.

Find L(6,10) mod 1010.