Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs
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| Format: | Preprint |
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2023
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| _version_ | 1866917647400042496 |
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| author | Chen, Xi Sokal, Alan D. |
| author_facet | Chen, Xi Sokal, Alan D. |
| contents | We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\mathbfϕ)$ in infinitely many indeterminates that give a weight $y$ to each improper edge and a weight $m! \, ϕ_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\mathbfϕ$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_03999 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs Chen, Xi Sokal, Alan D. Combinatorics Classical Analysis and ODEs 05A15 (Primary), 05A19, 05A20, 05C05, 05C30, 15B05, 15B36, 15B48, 30E05, 44A60 (Secondary) We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\mathbfϕ)$ in infinitely many indeterminates that give a weight $y$ to each improper edge and a weight $m! \, ϕ_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\mathbfϕ$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays. |
| title | Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs |
| topic | Combinatorics Classical Analysis and ODEs 05A15 (Primary), 05A19, 05A20, 05C05, 05C30, 15B05, 15B36, 15B48, 30E05, 44A60 (Secondary) |
| url | https://arxiv.org/abs/2302.03999 |