Searcharxiv⌕ Search

arXiv subjects

Alan D. Sokal

Publications and source records attributed to Alan D. Sokal.

At least 19 recordsLinked to original sources
arXiv2025

Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness

Given a lower-triangular matrix of real numbers, one can ask the following four total-positivity questions: total positivity of the triangle itself; total positivity of its row-reversal; Toeplitz-total positivity of its row sequences (equivalent to negative-real-rootedness of the row-generating polynomials); and coefficientwise Hankel-total positivity of the sequence of row-generating polynomials. In this paper, we introduce two infinite families of lower-triangular matrices generalising the Stirling cycle and subset triangles, parametrised by an integer $r \ge 1$; we call these the $r$th-order Stirling cycle and subset numbers. We then ask the foregoing four questions for each of these triangles, leading us to several conjectures. We then prove some of these conjectures for the case $r=2$.

math.CO↗
arXiv2024

Thron-type continued fractions (T-fractions) for some classes of increasing trees

We introduce some classes of increasing labeled and multilabeled trees, and we show that these trees provide combinatorial interpretations for certain Thron-type continued fractions with coefficients that are quasi-affine of period 2. Our proofs are based on bijections from trees to labeled Motzkin or Schröder paths; these bijections extend the well-known bijection of Françon--Viennot (1979) interpreted in terms of increasing binary trees. This work can also be viewed as a sequel to the recent work of Elvey Price and Sokal (2020), where they provide combinatorial interpretations for Thron-type continued fractions with coefficients that are affine. Towards the end of the paper, we conjecture an equidistribution of vincular patterns on permutations.

math.CO↗
arXiv2024

Continued fractions for cycle-alternating permutations

A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index $i$ is either a cycle valley ($σ^{-1}(i)>i<σ(i)$) or a cycle peak ($σ^{-1}(i) σ(i)$). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes--Rogers and Jacobi--Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs.

math.CO↗
arXiv2024

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

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.

math.CO↗
arXiv2024

A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle

We show that very simple continued fractions can be obtained for the ordinary generating functions enumerating permutations or D-permutations with a large number of independent statistics, when each cycle is given a weight $-1$. The proof is based on a simple lemma relating the number of cycles modulo 2 to the numbers of fixed points, cycle peaks (or cycle valleys), and crossings.

math.CO↗
arXiv2023

Lattice paths and branched continued fractions. III. Generalizations of the Laguerre, rook and Lah polynomials

We introduce a triangular array $\widehat{\sf L}^{(α)}$ of 5-variable homogeneous polynomials that enumerate Laguerre digraphs (digraphs in which each vertex has out-degree 0 or 1 and in-degree 0 or 1) with separate weights for peaks, valleys, double ascents, double descents, and loops. These polynomials generalize the classical Laguerre polynomials as well as the rook and Lah polynomials. We show that this triangular array is totally positive and that the sequence of its row-generating polynomials is Hankel-totally positive, under suitable restrictions on the values given to the indeterminates. This implies, in particular, the coefficientwise Hankel-total positivity of the monic unsigned univariate Laguerre polyomials. Our proof uses the method of production matrices as applied to exponential Riordan arrays. Our main technical lemma concerns the total positivity of a large class of quadridiagonal production matrices; it generalizes the tridiagonal comparison theorem. In some cases these polynomials are given by a branched continued fraction. Our constructions are motivated in part by recurrences for the multiple orthogonal polynomials associated to weights based on modified Bessel functions of the first kind $I_α$.

math.CO↗
arXiv2022

A simple algorithm for expanding a power series as a continued fraction

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).

math.CO↗
arXiv2022

Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers

A D-permutation is a permutation of $[2n]$ satisfying $2k-1 \le σ(2k-1)$ and $2k \ge σ(2k)$ for all $k$; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.

math.CO↗
arXiv2022

Ergodicity of the Wang--Swendsen--Kotecký algorithm on several classes of lattices on the torus

We prove the ergodicity of the Wang--Swendsen--Kotecký (WSK) algorithm for the zero-temperature $q$-state Potts antiferromagnet on several classes of lattices on the torus. In particular, the WSK algorithm is ergodic for $q\ge 4$ on any quadrangulation of the torus of girth $\ge 4$. It is also ergodic for $q \ge 5$ (resp. $q \ge 3$) on any Eulerian triangulation of the torus such that one sublattice consists of degree-4 vertices while the other two sublattices induce a quadrangulation of girth $\ge 4$ (resp.~a bipartite quadrangulation) of the torus. These classes include many lattices of interest in statistical mechanics.

cond-mat.stat-mech↗
arXiv2022

Skier and loop-the-loop with friction

We solve analytically the differential equations for a skier on a circular hill and for a particle on a loop-the-loop track when the hill or track is endowed with a coefficient of kinetic friction $μ$. For each problem, we determine the exact "phase diagram" in the two-dimensional parameter plane.

physics.class-ph↗
arXiv2022

When does a hypergeometric function ${}_{p\!}F_q$ belong to the Laguerre--Pólya class $LP^+$?

I show that a hypergeometric function ${}_{p}F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;\,\cdot\,)$ with $p \le q$ belongs to the Laguerre--Pólya class $LP^+$ for arbitrarily large $b_{p+1},\ldots,b_q > 0$ if and only if, after a possible reordering, the differences $a_i - b_i$ are nonnegative integers. This result arises as an easy corollary of the case $p=q$ proven two decades ago by Ki and Kim. I also give explicit examples for the case ${}_{1}F_2$.

math.CA↗
arXiv2022

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

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 $m! \, ϕ_m$ for each vertex with $m$ proper children. We show that if the weight sequence $ϕ$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

math.CO↗
arXiv2022

Multiple Laguerre polynomials: Combinatorial model and Stieltjes moment representation

I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$.

math.CA↗