A combinatorial bijection on $k$-noncrossing partitions

TL;DR

This paper establishes a combinatorial proof of an Euler transformation identity relating noncrossing partitions and enhanced crossings for any integer k ≥ 2, generalizing known results for specific cases.

Contribution

It provides a new combinatorial bijection proof for the Euler transformation identity involving k-noncrossing partitions and enhanced crossings.

Findings

Proves the Euler transformation identity combinatorially for all k≥2.

Generalizes the special case of Motzkin and Catalan numbers.

Connects pattern avoidance in sequences with noncrossing partition enumeration.

Abstract

For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of weights, $t^\text{number of blocks}$, of partitions of $\{1,\ldots,m\}$ without $k$-crossings (resp.~enhanced $k$-crossings). The special $k=2$ and $t=1$ case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for $k=3$ and $t=1$, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.