English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study three combinatorial models for the lower-triangular matrix withentries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on thevertex set $[n+1]$, and one involving partial functional digraphs on the vertexset $[n]$. We show that this matrix is totally positive and that the sequenceof its row-generating polynomials is coefficientwise Hankel-totally positive.We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and properedges, and further to polynomials $t_{n,k}(y,\mathbf{\phi})$ in infinitely manyindeterminates that give a weight $y$ to each improper edge and a weight $m! \,\phi_m$ for each vertex with $m$ proper children. We show that if the weightsequence $\mathbf{\phi}$ is Toeplitz-totally positive, then the two foregoingtotal-positivity results continue to hold. Our proofs use production matricesand exponential Riordan arrays.

Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs