Gilmore, T;
(2021)
Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices.
The Electronic Journal of Combinatorics
, 28
(3)
10.37236/10465.
Preview |
Text
Gilmore_Trees, forests, and total positivity- I.q-trees andq-forests matrices_VoR.pdf - Published Version Download (1MB) | Preview |
Abstract
We consider matrices with entries that are polynomials inqarising from naturalq-generalisations of two well-known formulas that count: forests onnvertices withkcomponents; and rooted labelled trees onn+ 1 vertices wherekchildren of the rootare lower-numbered than the root. We give a combinatorial interpretation of thecorresponding statistic on forests and trees and show, via the construction of vari-ous planar networks and the Lindstr ̈om-Gessel-Viennot lemma, that these matricesare coefficientwise totally positive. We also exhibit generalisations of the entriesof these matrices to polynomials ineightindeterminates, and present some conjec-tures concerning the coefficientwise Hankel-total positivity of their row-generatingpolynomials.
| Type: | Article |
|---|---|
| Title: | Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.37236/10465 |
| Publisher version: | https://doi.org/10.37236/10465 |
| Language: | English |
| Additional information: | ©The author. Released under the CC BY-ND license (International 4.0). |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10138471 |
Archive Staff Only
 |
View Item |
