Integer Partitions with Fixed Subsums

  • Yu. Yakubovich


Given two positive integers $m\le n$, we consider the set of partitions $\lambda=(\lambda_1,\dots,\lambda_\ell,0,\dots)$, $\lambda_1\ge\lambda_2\ge\dots$, of $n$ such that the sum of its parts over a fixed increasing subsequence $(a_j)$ is $m$: $\lambda_{a_1}+\lambda_{a_2}+\dots=m$. We show that the number of such partitions does not depend on $n$ if $m$ is either constant and small relatively to $n$ or depend on $n$ but is close to its largest possible value: $n-ma_1=k$ for fixed $k$ (in the latter case some additional requirements on the sequence $(a_j)$ are needed). This number is equal to the number of so-called colored partitions of $m$ (respectively $k$). It is proved by constructing bijections between these objects.

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