A combinatorial proof of a symmetry for a refinement of the Narayana numbers

TL;DR

The paper addresses refining Narayana numbers by jointly counting Dyck paths according to the numbers of $UD$-factors and $UUD$-factors, establishing the key symmetry $w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$. It develops a purely combinatorial proof built on extended peaks, necklaces, and the cycle lemma, and links these refined counts to Narayana numbers and Callan's generalized Narayana numbers. It also delivers a closed-form expression for $w_{n,k,m}$, generalizes the symmetry to broader factor families, and derives Catalan and generalized Narayana identities, together with real-rootedness and gamma-positivity results for associated polynomials $W_{n,k}(t)$. These contributions deepen understanding of Dyck-path statistics, reveal new symmetries, and connect classical Narayana structures to generalized variants, with polynomial consequences that support unimodality and symmetry properties. The work thus provides a rich combinatorial framework and multiple avenues for further exploration in enumerative combinatorics of lattice paths and related objects.

Abstract

We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$, then $w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $w_{2k+1,k,m}$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $w_{n,k,m}$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $w_{n,k_{1},k_{2},\ldots , k_{r}}$ counting the semilength-$n$ Dyck paths with $k_{1}$ $UD$-factors, $k_{2}$ $UUD$-factors, $\ldots$ , and $k_{r}$ $U^{r}D$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$, including real-rootedness, $γ$-positivity, and a symmetric decomposition.

Figures (2)

• Figure 1: Dyck paths of semilength $5$ with $2$ peaks where peaks that are part of a $UUD$-factor are colored red.
• Figure 2: Example of the bijection given in Definition \ref{['dfn:bijection']} for $k = 4$ where the Lalanne-Kreweras involution Kreweras1970Lalanne1992 is used in Step (4).

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2: Cycle lemma dvoretzky-motzkin
• Corollary 2.3: of the cycle lemma
• proof