On Graphical Partitions with Restricted Parts
Gilead Levy
TL;DR
This paper studies how arithmetic restrictions on partition parts affect the prevalence of graphical partitions of an even integer $n$. It introduces a general restriction model via a function $\mu$ and analyzes the induced random partition ensemble $\mathcal{P}_\mu$ with a $q$-weighted measure, tracking the statistics $X_k$ and $Y_k$. Using a probabilistic framework with Poissonization, saddle-point methods, and Esseen-type bounds, the authors show the graphical-fraction among restricted partitions is $O(n^{-1/2})$, with the slowest decay when the restriction is linear. Moreover, they prove the existence of a minimal lower bound $l_n$ on the parts beyond which the restrictions' influence becomes negligible, with $l_n = \mu(\log n) - \log n$; in the binary-partition case this specializes to $l_n = \frac{n^{\log 2}}{2} + O(\log n)$. These results illuminate how multiple restrictions interact to shape graphical partitions and suggest avenues for tighter closed-form expressions and broader restriction types.
Abstract
We study the distributions of parts in random integer partitions subject to general arithmetic restrictions. In particular, we enumerate restricted graphical partitions of an even integer $n$ and identify the conditions under which the fraction of graphical partitions, relative to all restricted partitions, is maximal. We prove that this maximal fraction is asymptotically $O(n^{-1/2})$. Furthermore, for any set of arithmetic restrictions, we establish the existence of a minimal lower bound on the parts beyond which the influence of these restrictions on the fraction of graphical partitions becomes negligible; in this regime, the fraction depends primarily on the choice of this lower bound. We highlight a key example of partitions restricted to powers of 2, where the critical lower bound is found to be $\frac{1}{2}n^{\log2}+O(\log n)$.