arXiv subjects

Michaela A. Polley

Publications and source records attributed to Michaela A. Polley.

3 recordsLinked to original sources

arXiv2026

Enumerating separable derangements

We give a polynomial-time algorithm to compute the number $b_n$ of separable derangements of $[n]$. This algorithm is based on a generating function technique which tracks permutations along with their occupied diagonals, where each permutation is counted once for every such diagonal. We provide bounds for the proportion of separable permutations which are derangements, show that $b_n$ and the large Schröder numbers have the same exponential growth constant $(3 + 2 \sqrt{2})$, and use the first 3000 terms to conjecture more explicit asymptotic behavior. This partially answers several questions about separable derangements recently posed by Vatter.

math.CO↗

arXiv2025

Patterns in rectangulations. Part I: $\top$-like patterns, inversion sequence classes $I(010, 101, 120, 201)$ and $I(011, 201)$, and rushed Dyck paths

We initiate a systematic study of pattern avoidance in rectangulations. We give a formal definition of such patterns and investigate rectangulations that avoid $\top$-like patterns - the pattern $\top$ and its rotations. For every $L \subseteq \{\top, \, \vdash, \, \bot, \, \dashv \}$ we enumerate $L$-avoiding rectangulations, both weak and strong. In particular, we show $\top$-avoiding weak rectangulations are enumerated by Catalan numbers and construct bijections to several Catalan structures. Then, we prove that $\top$-avoiding strong rectangulations are in bijection with several classes of inversion sequences, among them $I(010,101,120,201)$ and $I(011,201)$ - which leads to a solution of the conjecture that these classes are Wilf-equivalent. Finally, we show that $\{\top, \bot\}$-avoiding strong rectangulations are in bijection with recently introduced rushed Dyck paths.

math.CO↗

arXiv2022

The 334-Triangle Graph of $SL_3({\mathbb Z})$

Long, Reid, and Thistlewaite have shown that some groups generated by representations of the $Δ334$ triangle group in $SL_3({\mathbb Z})$ are thin, while the status of others is unknown. In this paper we take a new approach: for each group we introduce a new graph that captures information about representations of $Δ334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for $SL_3({\mathbb Z}/2{\mathbb Z})$ to show that the chromatic number of the graph for $SL_3({\mathbb Z})$ is at most eight. By generating a portion of the graph for $SL_3({\mathbb Z})$ we show its chromatic number is at least four; we conjecture it is equal to four.

math.CO↗