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Equidistributions of mahonian statistics over pattern avoiding permutations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-2305-9764
2018 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 25, no 1, article id P1.7Article in journal (Refereed) Published
Abstract [en]

A Mahonian d-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern functions of length at most d. Babson and Ste- ingrímsson classified all Mahonian 3-functions up to trivial bijections and identified many of them with well-known Mahonian statistics in the literature. We prove a host of Mahonian 3-function equidistributions over permutations in Sn avoiding a single classical pattern in S3. Tools used include block decomposition, Dyck paths and generating functions.

Place, publisher, year, edition, pages
Australian National University Press, 2018. Vol. 25, no 1, article id P1.7
Keywords [en]
Dyck path statistic, Equidistribution, Mahonian statistic, Pattern avoidance, Polyomino, St-Wilf equivalence
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-222257ISI: 000432155700008Scopus ID: 2-s2.0-85040954000OAI: oai:DiVA.org:kth-222257DiVA, id: diva2:1180173
Note

QC 20180205

Available from: 2018-02-05 Created: 2018-02-05 Last updated: 2019-05-10Bibliographically approved
In thesis
1. Combinatorics and zeros of multivariate polynomials
Open this publication in new window or tab >>Combinatorics and zeros of multivariate polynomials
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate $d$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. p. 42
Series
TRITA-SCI-FOU ; 2019:33
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-251303 (URN)978-91-7873-210-4 (ISBN)
Public defence
2019-05-24, D3, Lindstedsvägen 5, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20190510

Available from: 2019-05-10 Created: 2019-05-09 Last updated: 2019-05-10Bibliographically approved

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