Extending Fine sequences: a link with forbidden patterns

TL;DR

This paper introduces a bivariate generalization of Fine sequences, providing enumeration formulas, generating trees, and bijections with permutation classes, thereby deepening the combinatorial understanding of forbidden patterns.

Contribution

It extends Fine sequences to a bivariate setting, offers new enumeration formulas, and establishes bijections with permutation classes characterized by forbidden subsequences.

Findings

Bijection between 321-avoiding derangements and Fine sequences

Enumeration formulas for the generalized relations

Generating trees for the new combinatorial structures

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.