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Turkey NMO 1999, P-3, the number of the functions

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December 23, 2010
functioncombinatorics proposedcombinatorics

Problem Statement

For any two positive integers nn and pp, prove that there are exactly (p+1)n+1−pn+1{{(p+1)}^{n+1}}-{{p}^{n+1}} functions f:{1,2,...,n}→{−p,−p+1,−p+2,....,p−1,p}f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\} such that ∣f(i)−f(j)∣≤p\left| f(i)-f(j) \right|\le p for all i,j∈{1,2,...,n}i,j\in \left\{ 1,2,...,n \right\}.
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