Some distributions in increasing and flattened permutations

TL;DR

This paper studies various statistical parameters of increasing and flattened permutations, providing generating functions and combinatorial correspondences to understand their distributions and popularity.

Contribution

It introduces exponential generating functions for multiple parameters and establishes bijections between permutations and simpler combinatorial objects.

Findings

Derived explicit exponential generating functions for permutation parameters.

Established bijections linking permutations to simpler combinatorial structures.

Analyzed the distribution and popularity of parameters like descents, valleys, and minima.

Abstract

We examine the distribution and popularity of different parameters (such as the number of descents, runs, valleys, peaks, right-to-left minima, and more) on the sets of increasing and flattened permutations. For each parameter, we provide an exponential generating function for its corresponding distribution and popularity.Additionally, we present one-to-one correspondences between these permutations and some classes of simpler combinatorial objects.