Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements

Arthur Rodelet--Causse, Lenny Tevlin

TL;DR

This work develops a comprehensive framework for q-analogs of the two-parameter super Catalan numbers, deriving q-versions of classical identities (Touchard, Koshy, Reed Dawson), establishing $q$-convolution formulas, and obtaining a generating function for $q$-Catalan numbers. It then introduces Narayana-type refinements and proves their $\gamma$-positivity in the classical and type B settings, with a compatible treatment for type D, and extends these refinements to $q$-Narayana numbers, including $q$-Kreweras and Le Jen-Shoo identities and positivity results. The paper also provides structural positivity results for the $q$-Narayana polynomials and puts forward conjectures on unimodality and $q$-log-concavity. Collectively, these results unify and extend the theory of q-analogs in the super Catalan family and illuminate new directions for Narayana refinements and their combinatorial interpretations.

Abstract

We begin by deriving a number of combinatorial identities satisfied by the $q$-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the $q$-super Catalan numbers. Next, we introduce some $q$-convolution identities involving q-central binomial and q-Catalan numbers and derive a generating function for $q$-Catalan numbers. Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the $γ$-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural $q$-analogs, prove their $q$-$γ$-positivity and prove some identities they satisfy, generalizing identities of Kreweras and Le Jen-Shoo. Using yet another identity, we prove that these refinements are positive integer polynomials in $q$.

$q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements

TL;DR

This work develops a comprehensive framework for q-analogs of the two-parameter super Catalan numbers, deriving q-versions of classical identities (Touchard, Koshy, Reed Dawson), establishing -convolution formulas, and obtaining a generating function for -Catalan numbers. It then introduces Narayana-type refinements and proves their -positivity in the classical and type B settings, with a compatible treatment for type D, and extends these refinements to -Narayana numbers, including -Kreweras and Le Jen-Shoo identities and positivity results. The paper also provides structural positivity results for the -Narayana polynomials and puts forward conjectures on unimodality and -log-concavity. Collectively, these results unify and extend the theory of q-analogs in the super Catalan family and illuminate new directions for Narayana refinements and their combinatorial interpretations.

Abstract

We begin by deriving a number of combinatorial identities satisfied by the -super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the -super Catalan numbers. Next, we introduce some -convolution identities involving q-central binomial and q-Catalan numbers and derive a generating function for -Catalan numbers. Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the -positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural -analogs, prove their --positivity and prove some identities they satisfy, generalizing identities of Kreweras and Le Jen-Shoo. Using yet another identity, we prove that these refinements are positive integer polynomials in .
Paper Structure (17 sections, 21 theorems, 106 equations, 2 figures)

This paper contains 17 sections, 21 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $q$-super Catalan Identities
  3. A $q$-Touchard identity for the $q$-super Catalan numbers
  4. A $q$-Koshy identity for the $q$-super Catalan Numbers
  5. A $q$-Reed Dawson identity for $q$-super Catalan numbers
  6. Convolution formulas
  7. Generating function for $q$-Catalan numbers
  8. $\gamma$-positivity and Narayana refinements
  9. Type B noncrossing partitions
  10. Gamma-positivity of type B ($m=0$) Narayana polynomials
  11. The type D case
  12. A $q$-analog of the $m$-Narayana numbers
  13. The $q$-Kreweras Identity for $m$-Narayana numbers
  14. The Le Jen-Shoo identity for $m$-Narayana numbers
  15. The positivity of the $q$-$m$-Narayana polynomials
  16. ...and 2 more sections

Key Result

Proposition 1

Let $m$ and $n$ be nonnegative integers. We have the following formula that is the $q$-analog of the generalization of the Touchard identity to all super Catalan numbersHere and below we mark in $\textcolor{red}{red}$ the adjustments needed in the identities to accommodate the 'super' parameter $m$.

Figures (2)

  • Figure 1: The circular representation of the noncrossing partition of type B $(1,-3,-6),(-1,3,6),(2,-2),(4),(-4),(5),(-5)$.
  • Figure 2: Illustration of the bijection $\phi$ on an example in $NC^B(6)$.

Theorems & Definitions (43)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • Proposition 4
  • proof
  • ...and 33 more