[Paper Review] On super Catalan polynomials

Emily Allen, Irina Gheorghiciuc|arXiv (Cornell University)|Mar 20, 2014

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper introduces a q-analog of the super Catalan numbers, defined as $ S_q(m,n) = \frac{[2m]!_q[2n]!_q}{[m]!_q[n]!_q[m+n]!_q} $, which generalizes both the classical super Catalan numbers and F"urlinger-Hofbauer's $ q $-Catalan numbers. The key contribution is a combinatorial interpretation of $ T_q(2,n) = S_q(2,n)/(1+q^n) $ via a signed major index generating function over a specific class of ballot paths, establishing unimodality and a $ q $-analog of the super Catalan recurrence.

ABSTRACT

We present a $q$-analog of the super Catalan number $(2m)!(2n)!/2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(λ)$, due to Fürlinger and Hofbauer, for $λ=0$ and $λ=1$. We give a combinatorial interpretation for this analog when $m=2$.

Motivation & Objective

• To extend the $ q $-Catalan numbers to a broader class of polynomials that generalize super Catalan numbers.
• To provide a combinatorial interpretation of the $ q $-analog $ T_q(2,n) $, defined as $ S_q(2,n)/(1+q^n) $.
• To establish a $ q $-analog of the super Catalan recurrence $ 4T(m,n) = T(m+1,n) + T(m,n+1) $.
• To prove that $ S_q(m,n) $ is a polynomial with nonnegative integer coefficients and investigate its unimodality.

Proposed method

• Define the $ q $-analog $ S_q(m,n) $ using $ q $-factorials and $ q $-binomial coefficients.
• Introduce the $ q $-Ballot number $ B_q(n,r) $ as a generating function over paths in $ \mathcal{B}(n,r) $, with a signed major index statistic.
• Construct a bijection $ \psi $ between paths of height > $ 2r-1 $ and paths in $ \mathfrak{S}(n+r,n-r-1) $, preserving major index.
• Define a map $ g $ from a subset of ballot paths $ \mathcal{B}^{**}(n,2) $ to the set of Dyck paths $ \mathcal{C}_n \setminus \Omega_n $, preserving the difference $ \operatorname{maj} - \operatorname{des} $ up to a shift.
• Use the bijection $ g $ to relate $ B_q(n,1) $ and $ B_q^*(n,2) $, leading to the identity $ q^2 B_q(n,1) - B_q^*(n,2) = q^2 \sum_{\pi \in \Omega_n} q^{\operatorname{maj}(\pi) - \operatorname{des}(\pi)} $.
• Leverage the $ q $-analog of the super Catalan recurrence to derive identities involving $ T_q(m,n) $, including $ (1+q^n)(1+q^{n-m})T_q(m,n) = q^{n-m}T_q(n,m+1) + T_q(m,n+1) $.

Experimental results

Research questions

• RQ1 Can a $ q $-analog of the super Catalan numbers be constructed that generalizes both the classical super Catalan numbers and F"urlinger-Hofbauer's $ q $-Catalan numbers?
• RQ2 Is there a combinatorial interpretation of $ T_q(2,n) = S_q(2,n)/(1+q^n) $ in terms of lattice paths with signed major index statistics?
• RQ3 Does the $ q $-analog $ S_q(m,n) $ satisfy a $ q $-analog of the super Catalan recurrence $ 4T(m,n) = T(m+1,n) + T(m,n+1) $?
• RQ4 Can the unimodality of $ S_q(m,n) $ be established via combinatorial means?
• RQ5 What is the precise relationship between the generating function of $ T_q(2,n) $ and the set of Dyck paths with restricted height and descent structure?

Key findings

• The $ q $-analog $ S_q(m,n) $ is a polynomial with nonnegative integer coefficients, as established by Warnaar and Zudilin and confirmed via combinatorial construction.
• The polynomial $ T_q(2,n) = S_q(2,n)/(1+q^n) $ admits a combinatorial interpretation as $ \sum_{\pi \in \mathcal{B}^{**}(n,2)} q^{\operatorname{maj}(\pi) - \operatorname{des}(\pi)} $, where $ \mathcal{B}^{**}(n,2) $ is a specific class of ballot paths.
• The map $ g $ provides a bijection between $ \mathcal{B}^{**}(n,2) $ and $ \mathcal{C}_n \setminus \Omega_n $, such that $ \operatorname{maj}(\pi) - \operatorname{des}(\pi) = \operatorname{maj}(g(\pi)) - \operatorname{des}(g(\pi)) + 2 $, leading to the identity $ q^2 B_q(n,1) - B_q^*(n,2) = q^2 \sum_{\pi \in \Omega_n} q^{\operatorname{maj}(\pi) - \operatorname{des}(\pi)} $.
• The $ q $-analog of the super Catalan recurrence holds: $ (1+q^n)(1+q^{n-m})T_q(m,n) = q^{n-m}T_q(n,m+1) + T_q(m,n+1) $, generalizing the classical identity.
• For $ m=1 $, $ T_q(1,n) = \sum_{\pi \in \mathcal{C}_n} q^{\operatorname{maj}(\pi) - \operatorname{des}(\pi)} $, which matches the F"urlinger-Hofbauer $ q $-Catalan number $ c_n(0) $, and $ T_q(n,1) = \sum_{\pi \in \mathcal{C}_n} q^{\operatorname{maj}(\pi)} = c_n(1) $.
• The construction confirms that $ T_q(2,n) $ is a polynomial with nonnegative integer coefficients and provides a step toward proving unimodality of $ S_q(m,n) $.