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Putnam 2015 B5

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December 6, 2015
PutnamPutnam 2015Putnam combinatorics

Problem Statement

Let PnP_n be the number of permutations π\pi of {1,2,…,n}\{1,2,\dots,n\} such that ∣i−j∣=1 implies ∣π(i)−π(j)∣≤2|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2 for all i,ji,j in {1,2,…,n}.\{1,2,\dots,n\}. Show that for n≥2,n\ge 2, the quantity Pn+5−Pn+4−Pn+3+PnP_{n+5}-P_{n+4}-P_{n+3}+P_n does not depend on n,n, and find its value.
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