[Paper Review] Coefficientwise total positivity of some matrices defined by linear recurrences

Xi Chen, Bishal Deb+3 more|arXiv (Cornell University)|Dec 7, 2020

Advanced Combinatorial Mathematics5 references4 citations

TL;DR

This paper introduces a six-parameter lower-triangular matrix of polynomials defined by a linear recurrence, conjectured to be coefficientwise totally positive. The authors prove coefficientwise total positivity for a special case, $`\bm{T}(a,c,0,e,0,0)$, which generalizes the reversed Stirling subset triangle and provides a unified framework for understanding total positivity in combinatorial matrices like the Eulerian and Stirling triangles.

ABSTRACT

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of $T(a,c,0,e,0,0)$, which includes the reversed Stirling subset triangle.

Motivation & Objective

• To establish coefficientwise total positivity for a broad class of combinatorial matrices defined by linear recurrences.
• To provide a unified framework that includes the clean Eulerian triangle and the reversed Stirling subset triangle as special cases.
• To resolve long-standing conjectures on total positivity for the clean Eulerian and reversed Stirling subset triangles by proving a special case of a more general conjecture.
• To extend the notion of total positivity to matrices with polynomial entries using coefficientwise ordering.
• To prove a bijection between set partitions and lattice paths that supports the structural proof of total positivity in the special case.

Proposed method

• Define a lower-triangular matrix $\bm{T}(a,c,d,e,f,g)$ with entries in $\mathbb{Z}[a,c,d,e,f,g]$ via the recurrence $T(n,k) = [a(n-k)+c]T(n-1,k-1) + (dk+e)T(n-1,k) + [f(n-k)+g]T(n-1,k+1)$, with $T(0,k) = \delta_{k0}$.
• Introduce coefficientwise total positivity: a matrix with polynomial entries is coefficientwise totally positive if all its minors are polynomials with nonnegative coefficients.
• Prove that $\bm{T}(a,c,0,e,0,0)$ is coefficientwise totally positive using a combinatorial bijection between set partitions and lattice paths.
• Construct a map $\Phi_{n,k}$ from the set of partitions $\Pi_{n+1,n-k+1}$ to lattice paths $\mathsf{P}_{n,k}$, showing it is a bijection via recursive insertion of elements into blocks.
• Define path words $W(\pi)$ based on the order of element insertion and block structure, with letters $e_{i,l}, a_{i,0,l}, a_{i,j,l}$ corresponding to block behavior.
• Verify that the path words satisfy conditions ensuring they correspond to valid lattice paths, and that the inverse construction recovers the original partition.

Experimental results

Research questions

• RQ1 Is the general six-parameter matrix $\bm{T}(a,c,d,e,f,g)$ coefficientwise totally positive?
• RQ2 Does the special case $\bm{T}(a,c,0,e,0,0)$ exhibit coefficientwise total positivity, and can this be proven combinatorially?
• RQ3 Can the clean Eulerian triangle and the reversed Stirling subset triangle be recovered as specializations of this matrix, and does this imply their total positivity?
• RQ4 Is there a combinatorial interpretation of the minors of such matrices via lattice paths and set partitions?
• RQ5 Can the bijection between set partitions and lattice paths be used to prove total positivity for broader classes of recurrence-defined matrices?

Key findings

• The matrix $\bm{T}(a,c,0,e,0,0)$ is proven to be coefficientwise totally positive, establishing a key special case of the broader conjecture.
• The clean Eulerian triangle is recovered as the specialization $(a,c,d,e) = (1,1,1,1)$, and the reversed Stirling subset triangle as $(a,c,d,e) = (1,0,0,1)$, both lying within the proven framework.
• A bijection $\Phi_{n,k}: \Pi_{n+1,n-k+1} \to \mathsf{P}_{n,k}$ is constructed between set partitions and lattice paths, which underlies the proof of total positivity.
• The path words $W(\pi)$ associated with each partition are shown to satisfy precise combinatorial conditions that correspond to valid paths in the lattice path model.
• The proof relies on a recursive insertion algorithm that builds partitions block by block, tracking block status (finished, unfinished) and element order via the word $W(\pi)$.
• After specialization of path letters ($e_{i,l} \to e$, $a_{i,0,l} \to c$, $a_{i,j,l} \to a$), the recurrence matches $\bm{T}(a,c,0,e,0,0)$, confirming consistency with the original matrix.