eprintid: 10179955
rev_number: 11
eprint_status: archive
userid: 699
dir: disk0/10/17/99/55
datestamp: 2023-11-23 11:29:28
lastmod: 2023-11-23 11:29:28
status_changed: 2023-11-23 11:29:28
type: thesis
metadata_visibility: show
sword_depositor: 699
creators_name: Deb, Bishal
title: Enumerative combinatorics, continued fractions and total positivity
ispublished: unpub
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
note: Copyright © The Author 2023.  Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/).  Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms.  Access may initially be restricted at the author’s request.
abstract: Determining whether a given number is positive is a fundamental question in mathematics. This can sometimes be answered by showing that the number counts some collection of objects, and hence, must be positive. The work done in this dissertation is in the field of enumerative combinatorics, the branch of mathematics that deals with exact counting. We will consider several problems at the interface between
enumerative combinatorics, continued fractions and total positivity.

In our first contribution, we exhibit a lower-triangular matrix of polynomials in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. This generalises Brenti’s conjecture from 1996. We prove the coefficientwise total positivity of a three-variable case which includes the reversed Stirling subset triangle.

Our next contribution is the study of two sequences whose Stieltjes-type continued fraction coefficients grow quadratically; we study the Genocchi and median
Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations, a class of permutations of 2n, with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.

After this, we interpret the Foata–Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. Using this interpretation,
we obtain Jacobi-type continued fractions for multivariate polynomials enumerating permutations, and also Thron-type and Stieltjes-type continued fractions for multivariate polynomials enumerating D-permutations, in both cases including the counting of cycles. This enables us to prove some conjectured continued fractions due to Sokal–Zeng from 2022, and Randrianarivony–Zeng from 1996.

Finally, we introduce the higher-order Stirling cycle and subset numbers; these generalise the Stirling cycle and subset numbers, respectively. We introduce some
conjectures which involve different total-positivity questions for these triangular arrays and then answer some of them.
date: 2023-10-28
date_type: published
oa_status: green
full_text_type: other
thesis_class: doctoral_open
thesis_award: Ph.D
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 2100926
lyricists_name: Deb, Bishal
lyricists_id: BDEBX71
actors_name: Deb, Bishal
actors_id: BDEBX71
actors_role: owner
full_text_status: public
pages: 296
institution: UCL (University College London)
department: Mathematics
thesis_type: Doctoral
citation:        Deb, Bishal;      (2023)    Enumerative combinatorics, continued fractions and total positivity.                   Doctoral thesis  (Ph.D), UCL (University College London).     Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10179955/1/BishalDeb_PhDThesis_Final.pdf