تقرير
Counting pattern-avoiding permutations by big descents
| العنوان: | Counting pattern-avoiding permutations by big descents |
|---|---|
| المؤلفون: | Elizalde, Sergi, Rivera Jr., Johnny, Zhuang, Yan |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Primary 05A15, Secondary 05A05, 05A19 |
| الوصف: | A descent $k$ of a permutation $\pi=\pi_{1}\pi_{2}\dots\pi_{n}$ is called a big descent if $\pi_{k}>\pi_{k+1}+1$; denote the number of big descents of $\pi$ by $\operatorname{bdes}(\pi)$. We study the distribution of the $\operatorname{bdes}$ statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets $\Pi\subseteq\mathfrak{S}_{3}$ of size 1 and 2 into $\operatorname{bdes}$-Wilf equivalence classes, and we derive a formula for the distribution of big descents for each of these classes. Our methods include generating function techniques along with various bijections involving objects such as Dyck paths and binary words. Several future directions of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity. Comment: 36 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2408.15111 |
| رقم الأكسشن: | edsarx.2408.15111 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |