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Theorem 1: Let G be a directed acyclic graph with n designated origin and destination nodes, and let A be the n-by-n matrix whose (i, j)-entry is the number of paths from the ith origin to the jth destination. The following statements hold: (a) The number of n-paths is equal to the permanent of A. (b) If G is nonpermutable, the number of nonintersecting n-paths is equal to the determinant of A. (c) In general, even if G is not nonpermutable, the determinant of A equals Even(G) _ Odd(G), where Even(G) is the number of nonintersecting n-paths corresponding to even permutations and Odd(G) is the number of nonintersecting n-paths corresponding to odd permutations.

1Association Football and How to Play It

By John Cameron

An overview of the positions, tactics, and history of association football written by one of the game's early stars. John Cameron was a most interesting figure who played for both Everton and Tottenham, was a P.O.W. during the First World War and a mighty contributor to the organization of football among his fellow prisoners, and made his living as a journalist in later life. - Summary by Ben Adams