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$(q,t)$-Catalan numbers: gamma expansions, pattern avoidance and the $(-1)$-phenomenon
- Publication Year :
- 2018
-
Abstract
- The aim of this paper is two-fold. We first prove several new interpretations of a kind of $(q,t)$-Catalan numbers along with their corresponding $\gamma$-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain $(-1)$-phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their $q$-analogues utilizing the newly obtained $q$-$\gamma$-expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely $(2413,3142)$ and $(1342,2431)$, of length four, and consider such $(-1)$-phenomenon for these two subsets as well.<br />Comment: 36 pages, 3 figures, 4 tables
- Subjects :
- Mathematics - Combinatorics
05A05, 05A15, 05A19
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1805.08945
- Document Type :
- Working Paper