A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions

TL;DR

This paper introduces a simple combinatorial extension map that relates various types of k-noncrossing partitions, providing a new proof of their interconnectedness and resolving a recent conjecture.

Contribution

It presents a novel extension map that connects different classes of k-noncrossing partitions and offers a combinatorial proof of their binomial transform relationships.

Findings

The extension map preserves and shifts the crossing and nesting properties.

It establishes a direct combinatorial link between enhanced, classical, and 2-distant k-noncrossing partitions.

The work confirms a recent conjecture and generalizes previous identities.

Abstract

In this note, we give a simple extension map from partitions of subsets of [n] to partitions of [n+1], which sends $\delta$-distant k-crossings to $(\delta+1)$-distant k-crossings (and similarly for nestings). This map provides a combinatorial proof of the fact that the numbers of enhanced, classical, and 2-distant k-noncrossing partitions are each related to the next via the binomial transform. Our work resolves a recent conjecture of Zhicong Lin and generalizes earlier reduction identities for partitions.