Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations

TL;DR

This paper explores combinatorial structures related to Grand-Dyck paths, coloured noncrossing partitions, and signed pattern avoiding permutations, establishing new bijections and lattice isomorphisms to deepen understanding of their interrelations.

Contribution

It introduces new bijections between Grand-Dyck paths and signed pattern avoiding permutations, and transfers lattice structures to these combinatorial objects.

Findings

Established bijections between Grand-Dyck paths and signed pattern avoiding permutations

Transferred distributive lattice structures to coloured noncrossing partitions and permutations

Proved isomorphism with the Bruhat order on certain signed permutations

Abstract

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.