We count the pairs of walks between diagonally opposite corners of a given lattice rectangle by the number of points in which they intersect. We note that the number of such pairs with one intersection is twice the number with no intersection and we give a bijective proof of that fact. Some probabilistic variants of the problem are also investigated.
This note presents a simple, universal closed form for the powers of any square matrix. A diligent search of the internet gave no indication that the form is known.
Let M n, k r, s be the number of ordered pairs of paths in the plane, with unit steps E or N, that intersect k times in which the first path ends at the point (r, n&r) and the second path ends at the point (s, n&s). Let
Let $f_{n}= \sum_{i=0}^n {n \choose i}{ 2n-2i\choose n-i}$, $g_{n}= \sum_{i=1}^n {n\choose i}{2n-2i \choose n-i}$. Let $\{a_k\}_{k=1}$ be the set of all positive integers n, in increasing order, for which ${2n \choose n}$ is not divisible by 5, and let $\{b_k\}_{k=1}$ be the set of all positive integers n, in increasing order, for which $g_n$ is not divisible by 5. This note finds simple formulas for $a_k$, $b_k$, ${2n \choose n}$ mod 10, $ f_{n}$ mod 10, and $ g_{n}$ mod 10.
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