arXiv subjects

Elena L. Wang

Publications and source records attributed to Elena L. Wang.

4 recordsLinked to original sources

arXiv2026

A Grammatical Calculus for the Ramanujan Polynomials

The Ramanujan polynomials arise in three intertwined contexts. As remarked by BerndtEvans-Wilson, no combinatorial perspective seems to be alluded to in the original definition of Ramanujan. On a different stage, Dumont-Ramamonjisoa uncovered a combinatorial structure underneath an equation also considered by Ramanujan. Around the same time, Shor came up with the same construction as a refinement of the classical formula of Cayley for trees. We present a labeling scheme for rooted trees by employing an extra label marking improper edges. Harnessed by this grammar, we develop a grammatical calculus for the Ramanujan polynomials heavily relying on the constant properties. Moreover, we provide a grammatical formulation of a correspondence that leads to the recurrence relation due to Berndt-Evans-Wilson and Shor.

math.CO↗

arXiv2025

Enriched Cycle Structures and Roots of Permutations

This paper is concerned with a duality between $r$-regular permutations and $r$-cycle permutations, and a monotone property due to Bóna-McLennan-White on the probability $p_r(n)$ for a random permutation of $\{1,2,\ldots, n\}$ to have an $r$-th root, where $r$ is a prime. For $r=2$, the duality relates permutations with odd cycles to permutations with even cycles. To handle the general case where $r\geq 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles colored by one of the colors $1, 2, \ldots, r-1$. In this setup, we discover a bijection between $r$-regular permutations and enriched $r$-cycle permutations, which in turn yields a stronger version of an inequality of Bóna-McLennan-White. This leads to a fully combinatorial understanding of the monotone property, thereby answering their question. When $r$ is a prime power $q^l$, we further show that $p_r(n)$ is monotone. In the case that $n+1 \not\equiv 0 \pmod q$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.

math.CO↗

arXiv2025

On Ward Numbers and Increasing Schröder Trees

The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schröder trees} by verifying they satisfy Ward's recurrence. We construct a direct type-preserving bijection between total partition trees and increasing Schröder trees, complementing known type-preserving bijections to set partitions (including Chen's decomposition for increasing Schröder trees). Weighted generalizations extend these bijections to enriched increasing Schröder trees trees and Schröder trees trees, yielding new links to labeled rooted trees. Finally, we deduce a functional equation for weighted increasing Schröder trees, whose solution using Chen's decomposition leads to a combinatorial interpretation of a Lagrange inversion variant.

math.CO↗

arXiv2024

The Wide Band Cayley Continuants

The Cayley continuants are referred to the determinants of tridiagonal matrices in connection with the Sylvester continuants. Munarini-Torri found a striking combinatorial interpretation of the Cayley continuants in terms of the joint distribution of the number of odd cycles and the number of even cycles of permutations of $[n]=\{1,2,\ldots, n\}$. In view of a general setting, $r$-regular cycles (with length not divisible by $r$) and $r$-singular cycles (with length divisible by $r$) have been extensively studied largely related to roots of permutations. We introduce the wide band Cayley continuants as an extension of the original Cayley continuants, and we show that they can be interpreted in terms of the joint distribution of the number of $r$-regular cycles and the number of $r$-singular cycles over permutations of $[n]$.

math.CO↗