1986 USAMO Problems/Problem 5

Problem

By a partition $\pi$ of an integer $n\ge 1$ , we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$ , then the partitions $\pi$ are $1+1+1+1$ , $1+1+2$ , $1+3, 2+2$ , and $4$ ).

For any partition $\pi$ , define $A(\pi)$ to be the number of $1$ 's which appear in $\pi$ , and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$ . (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$ , then $A(\pi)=2$ and $B(\pi) = 3$ ).

Prove that, for any fixed $n$ , the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$ .

Solution

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1986 USAMO (Problems • Resources)
Preceded by Problem 4	Followed by Last Question
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1986 USAMO Problems/Problem 5

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