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{\bf Yu.~Yakubovich}
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{\bf Integer Partitions with Fixed Subsums}
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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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