{ "id": "1812.10738", "version": "v1", "published": "2018-12-27T15:02:22.000Z", "updated": "2018-12-27T15:02:22.000Z", "title": "Revisiting pattern avoidance and quasisymmetric functions", "authors": [ "Jonathan Bloom", "Bruce Sagan" ], "comment": "25 pages", "categories": [ "math.CO" ], "abstract": "Let S_n be the nth symmetric group. Given a set of permutations Pi we denote by S_n(Pi) the set of permutations in S_n which avoid Pi in the sense of pattern avoidance. Consider the generating function Q_n(Pi) = sum_pi F_{Des pi} where the sum is over all pi in S_n(Pi) and F_{Des pi} is the fundamental quasisymmetric function corresponding to the descent set of pi. Hamaker, Pawlowski, and Sagan introduced Q_n(Pi) and studied its properties, in particular, finding criteria for when this quasisymmetric function is symmetric or even Schur nonnegative for all n >= 0. The purpose of this paper is to continue their investigation answering some of their questions, proving one of their conjectures, as well as considering other natural questions about Q_n(Pi). In particular we look at Pi of small cardinality, superstandard hooks, partial shuffles, Knuth classes, and a stability property.", "revisions": [ { "version": "v1", "updated": "2018-12-27T15:02:22.000Z" } ], "analyses": { "subjects": [ "05E05", "05A05" ], "keywords": [ "revisiting pattern avoidance", "nth symmetric group", "fundamental quasisymmetric function", "descent set", "stability property" ], "note": { "typesetting": "TeX", "pages": 25, "language": "en", "license": "arXiv", "status": "editable" } } }