[Paper Review] Extending Fine sequences: a link with forbidden patterns

Olivier Guibert, Sylvain Pelat-Alloin|arXiv (Cornell University)|Jul 20, 2005

Advanced Combinatorial Mathematics17 references3 citations

TL;DR

This paper introduces a bivariate generalization of Fine sequences—nonsingular similarity relations—using generating trees to establish a new bijection between 321-avoiding derangements and Fine sequences. It further links two special cases of these generalized sequences to permutations avoiding specific forbidden patterns, providing fully bijective proofs via generating tree constructions and enumeration formulas.

ABSTRACT

We propose a natural, bivariate, generalization of the nonsingular similarity relations considered by T. Fine. We also provide an enumeration formulae and a generating tree for those relations. The latter allow us to give a new bijection between 321-avoiding derangements and Fine sequences. Moreover, we establish that two special cases are in a one-to-one correspondence with subsets of permutations characterized by forbidden subsequences on the symmetrical group. All our results are established using the technique of generating tree, thus giving entirely bijective proofs.

Motivation & Objective

• To generalize nonsingular similarity relations (Fine sequences) into a bivariate framework using generating trees.
• To establish a new bijective correspondence between 321-avoiding derangements and Fine sequences through generating tree structures.
• To connect special cases of generalized Fine sequences with permutations avoiding specific forbidden patterns (e.g., 321, 1324).
• To provide fully bijective proofs for enumeration formulas using generating tree techniques.
• To extend the combinatorial understanding of Fine numbers and their connections to Catalan paths and ballot numbers.

Proposed method

• Define generalized Fine sequences $F_n^{p,q}$ as words over $\mathbb{N}^n$ with constraints on initial segments and local increments: $\omega_{i+1} \leq \omega_i + 1$, and a condition on zeros followed by $12\ldots(q-1)$.
• Use generating trees to model the growth of generalized Fine sequences, with labeled nodes representing permutations or sequences and recursive rules for generating children.
• Establish bijections between generating trees of generalized Fine sequences and those of permutations avoiding specific patterns (e.g., $321$, $1324$, $3214$).
• Apply structural properties of forbidden patterns to define active sites and labeling rules in generating trees, ensuring bijective consistency.
• Prove that insertion of $n+1$ into valid positions preserves forbidden pattern avoidance and induces correct label transitions in the generating tree.
• Use the generating tree framework to derive enumeration formulas and confirm that the number of such sequences matches known sequences like Fine numbers and ballot numbers.

Experimental results

Research questions

• RQ1 How can nonsingular similarity relations (Fine sequences) be naturally generalized into a bivariate framework?
• RQ2 What is the connection between generalized Fine sequences and permutations avoiding the 321 pattern, particularly in the context of derangements?
• RQ3 Which special cases of generalized Fine sequences correspond to permutations avoiding specific forbidden patterns (e.g., $1324$, $3214$)?
• RQ4 Can generating trees be used to provide fully bijective proofs for enumeration formulas of these combinatorial objects?
• RQ5 How do structural constraints on the initial rise and primitive paths in Catalan paths relate to the generalized Fine sequence construction?

Key findings

• The paper establishes a new bijection between $321$-avoiding derangements and Fine sequences using generating trees, providing a bijective proof of their equinumerosity.
• Two special cases of generalized Fine sequences are shown to be in one-to-one correspondence with permutations avoiding the forbidden patterns $1324$ and $3214$, respectively.
• The generating tree for $S_n({\cal H}_3^{-1c})$ is defined with root $[A,2]$ and labeled transitions that correctly model active site behavior and pattern avoidance.
• The generating tree for $S_n({\cal H}_5)$ is constructed with labels $[P]$, $[A,t]$, and $[B,t]$, where insertion rules preserve forbidden pattern avoidance and induce correct label transitions.
• The enumeration formula for generalized Fine sequences $F_n^{p,q}$ is derived via generating tree growth, confirming that the number of such sequences matches known combinatorial sequences.
• The paper confirms that $2F_n + F_{n-1} = C_{n+1}$, a known identity for Fine numbers, through bijective means using generating trees.