Counting subwords in flattened permutations

TL;DR

This paper studies the distribution of various subword patterns in flattened permutations, providing explicit formulas and connections to classical polynomials, and establishing equidistribution results.

Contribution

It introduces a unified approach to derive explicit formulas for subword pattern distributions in flattened permutations, linking them to Eulerian and Chebyshev polynomials.

Findings

Explicit formulas for descents and ascents using Eulerian polynomials.

Formulas for 123- and 321-subwords expressed via Chebyshev polynomials.

Bijection showing equidistribution of descents in flattened and usual permutations.

Abstract

In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on $\mathcal{S}_n$ in the flattened sense. For the other four patterns in question, we develop a unified approach to obtain explicit formulas for the comparable distributions. We find that the formulas so obtained for 123- and 321-subwords can be expressed in terms of the Chebyshev polynomials of the second kind, while those for peaks and valleys are more related to the Eulerian polynomials. We also provide a bijection showing the equidistribution of descents in flattened permutations of a given length with big descents in permutations of the same length in the usual sense.