Trees, forests, and total positivity. I: \(q\)-trees and \(q\)-forests matrices

DOI10.37236/10465zbMath1473.05021arXiv2106.00656OpenAlexW3204722398MaRDI QIDQ820844

Tomack Gilmore

Publication date: 28 September 2021

Published in: The Electronic Journal of Combinatorics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2106.00656

zbMATH Keywords

forests of rooted labelled treestotal positive matrices

Mathematics Subject Classification ID

Trees (05C05) Exact enumeration problems, generating functions (05A15) Combinatorial identities, bijective combinatorics (05A19) (q)-calculus and related topics (05A30) Combinatorial inequalities (05A20) Positive matrices and their generalizations; cones of matrices (15B48) Signed and weighted graphs (05C22)

Related Items

Total positivity of some polynomial matrices that enumerate labeled trees and forests. I: Forests of rooted labeled trees

Uses Software

• OEIS

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials
• A recursive algorithm for trees and forests
• Production matrices and riordan arrays
• Derivatives of the tree function
• Bessel polynomials
• A refinement of Cayley's formula for trees
• Arboretum for a generalisation of Ramanujan polynomials
• Bijections for Cayley trees, spanning trees, and their q-analogues
• Generalized Schur-complements and a test for total positivity
• Elimination techniques and interpolation
• \(q\)-extension of identities of Abel-Rothe type
• Combinatorial aspects of continued fractions
• Total positivity and Neville elimination
• A unified combinatorial approach for \(q\)- (and \(p,q\)-) Stirling numbers
• A Ramanujan sequence that refines the Cayley formula for trees
• Forest volume decompositions and Abel-Cayley-Hurwitz multinomial expansions
• A bijective proof of the Shor recurrence
• The \((q, t)\)-Gaussian process
• On the Lambert \(w\) function
• A note on Abel polynomials and rooted labeled forests
• A bijective proof of a Touchard-Riordan formula
• A new proof of Cayley's formula for counting labeled trees
• Combinatorics and total positivity
• Ramanujan grammar and Cayley trees
• Some applications of the \(q\)-exponential formula
• Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity
• A context-free grammar for the Ramanujan-Shor polynomials
• A remark on the enumeration of rooted labeled trees
• Lattice paths and branched continued fractions. II: Multivariate Lah polynomials and Lah symmetric functions
• A generalization of the Ramanujan polynomials and plane trees
• A \(q\)-analogue of the Riordan group
• A \(q\)-analog of the exponential formula. (Reprint)
• Distribution of crossings, nestings and alignments of two edges in matchings and partitions
• Coefficientwise total positivity of some matrices defined by linear recurrences
• On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs
• On Cayley's Formula for Counting Trees
• The Distribution of Crossings of Chords Joining Pairs of 2n Points on a Circle
• Proofs from THE BOOK
• Cayley′s Formula: A Page From The Book
• Forests of labeled trees
• A New Class of Orthogonal Polynomials: The Bessel Polynomials
• Sur Un Problème De Configurations Et Sur Les Fractions Continues
• Preuves combinatoires des identites d'Abel
• Total positivity: tests and parametrizations.