[Paper Review] Restricted inversion sequences and enhanced $3$-noncrossing partitions

Zhicong Lin|arXiv (Cornell University)|Jun 22, 2017

Advanced Combinatorial Mathematics3 citations

TL;DR

This paper proves a conjecture by Yan and Martinez–Savage that inversion sequences avoiding weakly decreasing subsequences of length 3 are equinumerous with enhanced 3-noncrossing partitions. Using generating trees and the obstinate kernel method, the author establishes a functional equation and solves it via Lagrange inversion and Zeilberger's algorithm, confirming the equinumerosity and deriving a novel identity linking classical and enhanced 3-noncrossing partitions.

ABSTRACT

We prove a conjecture due independently to Yan and Martinez--Savage that asserts inversion sequences with no weakly decreasing subsequence of length $3$ and enhanced $3$-noncrossing partitions have the same cardinality. Our approach applies both the generating tree technique and the so-called obstinate kernel method developed by Bousquet-Mélou. One application of this equinumerosity is a discovery of an intriguing identity involving numbers of classical and enhanced $3$-noncrossing partitions.

Motivation & Objective

• To prove the conjecture that the number of inversion sequences avoiding weakly decreasing subsequences of length 3 equals the number of enhanced 3-noncrossing partitions.
• To establish a functional equation for the generating function of these restricted inversion sequences using generating tree techniques.
• To solve the resulting functional equation via the obstinate kernel method and Lagrange inversion, thereby confirming the conjecture.
• To derive a new identity connecting classical and enhanced 3-noncrossing partitions: $C_3(n+1) = \sum_{i=0}^n \binom{n}{i} E_3(i)$.
• To explore connections between inversion sequences, ascent sequences, and set partitions, and to propose extensions to $k$-noncrossing structures.

Proposed method

• Construct a generating tree for the set of $(\geq,\geq,-)$-avoiding inversion sequences using left-to-right maxima and remaining entries as state parameters.
• Define a two-parameter generating function $F(u,v)$ encoding inversion sequences by their parameters $(p,q)$, where $p = \alpha(e) - \beta(e)$ and $q = n - \alpha(e)$.
• Derive a functional equation (5.1) for $F(u,v)$ based on the rewriting rules of the generating tree: $F(u,v) = t u v + \frac{t u v}{v - u}(F(v,1) - F(u,1)) + \frac{t u v}{1 - u}(F(u,1) - F(u,u))$.
• Apply the obstinate kernel method to solve the functional equation, leveraging symmetry and kernel conditions to extract coefficients.
• Use Lagrange inversion and Zeilberger’s algorithm to compute the coefficients of the generating function and confirm the conjectured enumeration.
• Extend the method to analyze $\mathcal{AW}$-inversion sequences, deriving a recursive functional equation (5.2) that allows computation of $|\mathbf{I}_n(\mathcal{AW})|$ despite not solving the closed form.

Experimental results

Research questions

• RQ1 Do inversion sequences avoiding weakly decreasing subsequences of length 3 have the same cardinality as enhanced 3-noncrossing partitions?
• RQ2 Can a functional equation for the generating function of $(\geq,\geq,-)$-avoiding inversion sequences be derived using generating tree techniques?
• RQ3 Is the obstinate kernel method applicable to solve the resulting functional equation and yield a closed-form enumeration?
• RQ4 Does the equinumerosity imply a deeper combinatorial identity between classical and enhanced 3-noncrossing partitions?
• RQ5 Can similar techniques be applied to other classes of restricted inversion sequences, such as $\mathcal{AW}$-inversion sequences?

Key findings

• The conjecture that $|\mathbf{I}_n(\geq,\geq,-)| = E_3(n)$ is confirmed, establishing equinumerosity between $(\geq,\geq,-)$-avoiding inversion sequences and enhanced 3-noncrossing partitions.
• A new identity is derived: $C_3(n+1) = \sum_{i=0}^n \binom{n}{i} E_3(i)$, linking classical and enhanced 3-noncrossing partitions.
• The functional equation for the generating function of $(\geq,\geq,-)$-avoiding inversion sequences is solved using the obstinate kernel method, yielding the correct enumeration sequence.
• The generating tree approach successfully models the structure of restricted inversion sequences via parameters $p$ and $q$, enabling recursive construction.
• The method is extended to $\mathcal{AW}$-inversion sequences, yielding a recursive functional equation (5.2) that allows computation of $|\mathbf{I}_n(\mathcal{AW})|$ even without a closed form.
• Recent developments confirm the result via a bijective construction using 01-fillings of triangular shapes, providing a combinatorial proof of the conjecture.