Problem 289
Eulerian Cycles
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.
欧拉回路
记C(x,y)是穿过点(x, y)、(x, y+1)、(x+1, y)和(x+1, y+1)的圆。
对于正整数m和n，记E(m,n)为包含如下m·n个圆的系统：
{ C(x,y): 0 ≤ x < m, 0 ≤ y < n, x和y为整数 }
在E(m,n)上的一个欧拉回路是一条穿过每条弧一次且恰好一次的闭路径。
在E(m,n)上这样的路径有许多条，但我们只关心那些不自穿的路径：一条不自穿的路径在格点上只接触而不穿过其本身。
下图展示了E(3,3)和一条不自穿的欧拉回路。
记L(m,n)是E(m,n)上所有不自穿欧拉回路的数目。
举例来说，L(1,2) = 2，L(2,2) = 37，以及L(3,3) = 104290。
求L(6,10) mod 1010。