Krystian.Gajdzica

Affiliation: Jagiellonian University

Title Of Talk: Rectangle partitions generalizing integer partitions

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Abstract: The story of integer partitions goes back to Euler, who, among other things, discovered the generating function for the partition function $p(n)$: $$ \sum_{n=0}^\infty p(n)q^n=\prod_{n=1}^\infty\frac{1}{1-q^n}. $$ There is a wealth of ways to generalize the partition function. In this talk, we present a ``geometric'' extension of the partition function, and consider the number $p(m,n)$ of ways to partition a rectangle of size $m\times n$ into rectangular blocks with integer sides, where two partitions of the rectangle are considered the same if they consist of the same multiset of blocks (their geometric arrangement is neglected). We show some basic properties of $p(m,n)$ and derive the analog of Hardy-Ramanujan formula in the case of $p(2,n)$. Moreover, we also investigate the asymptotic behavior of the number of restricted partitions of a rectangle, where only blocks of special sizes can be used as parts.

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