Sokal, Alan D;
(2022)
Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees.
Monatshefte für Mathematik
10.1007/s00605-022-01687-0.
(In press).
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Abstract
We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight m!ϕm for each vertex with m proper children. We show that if the weight sequence ϕϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.
| Type: | Article |
|---|---|
| Title: | Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00605-022-01687-0 |
| Publisher version: | https://doi.org/10.1007/s00605-022-01687-0 |
| Language: | English |
| Additional information: | Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| UCL classification: | UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics UCL > Provost and Vice Provost Offices > UCL BEAMS UCL |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10144793 |
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