[Paper Review] A note on log-convexity of q-Catalan numbers

Lynne M. Butler, W. Patrick Flanigan|ArXiv.org|Jan 2, 2007

Advanced Combinatorial Mathematics2 references4 citations

TL;DR

This paper establishes a generalized log-convexity property for $q$-Catalan numbers using a combinatorial injection between lattice path pairs. It proves that $C_{k-1}(q)C_{\ ilde{\ell}+1}(q) - q^{\ell-k+1}C_k(q)C_\ell(q)$ has nonnegative coefficients for $k \leq \ell$, extending classical log-convexity to the $q$-analog setting via area-preserving path transformations.

ABSTRACT

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.

Motivation & Objective

• To establish a stronger, generalized form of log-convexity for $q$-Catalan numbers beyond the standard definition.
• To resolve the subtlety in defining $q$-log-convexity, where $P_{k-1}(q)P_{k+1}(q) - P_k(q)^2 \geq 0$ does not imply the full pairwise inequality for $\ell \geq k$.
• To provide a combinatorial proof using lattice paths and inversion numbers, leveraging an injection that preserves area (i.e., $q$-degree) across path pairs.
• To demonstrate that the standard $q$-Catalan polynomials satisfy a refined inequality involving a $q$-factor that accounts for path size differences.

Proposed method

• Define $C_n(q)$ as the generating function for inversion numbers over lattice permutations with $n$ 1s and $n$ 2s, visualized as Dyck paths.
• Construct an injection $\varphi: \mathcal{P}_k \times \mathcal{P}_\ell \to \mathcal{P}_{k-1} \times \mathcal{P}_{\ell+1}$ based on the first meeting point of two lattice paths.
• Use the path decomposition $\pi = \pi_L \pi_R$, $\sigma = \sigma_L \sigma_R$ at the meeting index $i$, where $m_2(\pi_L) = m_2(\sigma_L) + 1$.
• Apply the inversion number formula: $\text{inv}(\tau) = \text{inv}(\tau_L) + \text{inv}(\tau_R) + (m_2\tau_L)(m_1\tau_R)$, to compare total inversion counts.
• Show that $\text{inv}(\nu) + \text{inv}(\omega) = \text{inv}(\pi) + \text{inv}(\sigma) + (\ell - k + 1)$, proving the $q$-factor shift.
• Visualize the transformation as swapping path segments at the first intersection, preserving the area (exponent of $q$) difference.

Experimental results

Research questions

• RQ1 Does the standard $q$-log-convexity definition ($P_{k-1}(q)P_{k+1}(q) - P_k(q)^2 \geq 0$) imply the full pairwise inequality $P_{k-1}(q)P_{\ell+1}(q) - q^{\ell-k+1}P_k(q)P_\ell(q) \geq 0$ for $\ell \geq k$?
• RQ2 Can a combinatorial injection be constructed to prove a refined log-convexity inequality for $q$-Catalan numbers?
• RQ3 What is the correct $q$-factor to balance the degree difference when comparing $C_{k-1}(q)C_{\ell+1}(q)$ and $C_k(q)C_\ell(q)$?
• RQ4 Why does the naive $q$-log-convexity fail for $q$-Catalan numbers, and what structural property enables the generalized inequality?

Key findings

• The expression $C_{k-1}(q)C_{\ell+1}(q) - q^{\ell-k+1}C_k(q)C_\ell(q)$ has nonnegative coefficients for all $k \leq \ell$, generalizing classical log-convexity to the $q$-setting.
• The injection $\varphi$ maps pairs of lattice paths $(\pi, \sigma)$ to $(\nu, \omega)$ such that the total $q$-degree increases by exactly $\ell - k + 1$.
• The proof relies on the path intersection point where $\pi_L$ first has one more 2 than $\sigma_L$, enabling a consistent decomposition and area accounting.
• The corollary extends the result to $C_{k-r}(q)C_{\ell+r}(q) - q^{r(\ell-k+r)}C_k(q)C_\ell(q)$, with the exponent $r(\ell-k+r)$ matching the degree of the product terms.
• The example confirms the degree equality: $\deg(q^{2(3)}C_6(q)C_7(q)) = \deg(C_4(q)C_9(q)) = 51$, and the injection preserves this under the transformation.
• The transformation is visualized as swapping path segments at the first meeting point in a shared grid, with the $q$-factor corresponding to the area of a rectangle in the lower-right quadrant.