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Combinatorics of stable polynomials and correlation inequalities
Stockholm University, Faculty of Science, Department of Mathematics.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains five papers divided into two parts. In the first part, Papers I-IV, we study polynomials within the field of combinatorics. Here we study combinatorial properties as well as the zero distribution of the polynomials in question. The second part consists of Paper V, where we study correlating events in randomly oriented graphs.

In Paper I we give a new combinatorial interpretation of the stationary distribution of the partially asymmetric exclusion process in terms of colored permutations and decorated alternative trees. We also find a connection between the corresponding multivariate partition functions and the multivariate Eulerian polynomials for r-colored permutations.

In Paper II we study a multivariate refinement of P-Eulerian polynomials. We show that this refinement is stable (i.e., non-vanishing whenever the imaginary parts of its variables are all positive) for a large class of labeled posets.

In Paper III we use the technique of compatible polynomials to prove that the local h-polynomial of the rth edgewise subdivision of the (n - 1)-dimensional simplex 2V has only real zeros. We generalize the result and study matrices with interlacing preserving properties.

In Paper IV we introduce s-lecture hall partitions for labeled posets. We provide generating functions as well as establish a connection between statistics on wreath products and statistics on lecture hall partitions for posets.

In Paper V we prove that the events {s → a} (that there exists a directed path from s to a) and {t → b} are positively correlated in a random tournament for distinct vertices as, b, t ∈ Kn. We also discuss the correlation between the same events in two random graphs with random orientation.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2016. , 32 p.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-128584ISBN: 978-91-7649-375-5 (print)OAI: oai:DiVA.org:su-128584DiVA: diva2:915712
Public defence
2016-05-27, room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript.

Available from: 2016-05-02 Created: 2016-03-30 Last updated: 2017-03-07Bibliographically approved
List of papers
1. Multivariate Eulerian Polynomials and Exclusion Processes
Open this publication in new window or tab >>Multivariate Eulerian Polynomials and Exclusion Processes
2016 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 25, no 4, 486-499 p.Article in journal (Refereed) Published
Abstract [en]

We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-128340 (URN)10.1017/S0963548316000031 (DOI)
Available from: 2016-03-23 Created: 2016-03-23 Last updated: 2017-11-30Bibliographically approved
2. Multivariate P-Eulerian polynomials
Open this publication in new window or tab >>Multivariate P-Eulerian polynomials
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The P-Eulerian polynomial counts the linear extensions of a labeled partially ordered set, P, by their number of descents. It is known that the P-Eulerian polynomials are real-rooted for various classes of posets P. The purpose of this paper is to extend these results to polynomials in several variables. To this end we study multivariate extensions of P-Eulerian polynomials and prove that for certain posets these polynomials are stable, i.e., non-vanishing whenever all variables are in the upper half-plane of the complex plane. A natural setting for our proofs is the Malvenuto-Reutenauer algebra of permutations (or the algebra of free quasi-symmetric functions). In the process we identify an algebra on Dyck paths, which to our knowledge has not been studied before.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-128347 (URN)
Available from: 2016-03-23 Created: 2016-03-23 Last updated: 2016-04-22Bibliographically approved
3. Compatible polynomials and edgewise subdivisions
Open this publication in new window or tab >>Compatible polynomials and edgewise subdivisions
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-128344 (URN)
Available from: 2016-03-23 Created: 2016-03-23 Last updated: 2016-04-22Bibliographically approved
4. Lecture hall P-partitions
Open this publication in new window or tab >>Lecture hall P-partitions
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-128346 (URN)
Available from: 2016-03-23 Created: 2016-03-23 Last updated: 2016-04-22Bibliographically approved
5. CORRELATION OF PATHS BETWEEN DISTINCT VERTICES IN A RANDOMLY ORIENTED GRAPH
Open this publication in new window or tab >>CORRELATION OF PATHS BETWEEN DISTINCT VERTICES IN A RANDOMLY ORIENTED GRAPH
2015 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 116, no 2, 287-300 p.Article in journal (Refereed) Published
Abstract [en]

We prove that in a random tournament the events {s -> a} (meaning that there is a directed path from s to a) and {t -> b} are positively correlated, for distinct vertices a, s, b,t is an element of K-n. It is also proven that the correlation between the events {s -> a} and {t -> b} in the random graphs G(n, p) and G(n, m) with random orientation is positive for every fixed p > 0 and sufficiently large n (with m = Is left perpendicularp((n)(2))left perpendicular). We conjecture it to be positive for all p and all n. An exact recursion for P({s -> a} boolean AND {t -> b} in G(n, p) is given.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-120489 (URN)000358751500006 ()
Available from: 2015-09-10 Created: 2015-09-10 Last updated: 2017-12-04Bibliographically approved

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