# Permutations of type $B$ with fixed number of descents and minus signs

## TL;DR

This paper investigates the enumeration of type B permutations with fixed descents and minus signs, revealing integrality properties and offering combinatorial interpretations for these counts.

## Contribution

It introduces a new array of numbers counting type B permutations with specific features and proves their integrality, along with providing combinatorial interpretations.

## Key findings

- Proves that B(n,k,j)/C(n,j) is an integer.
- Provides two combinatorial interpretations for B(n,k,j).
- Analyzes the structure of permutations with fixed descents and minus signs.

## Abstract

We study three dimensional array of numbers $B(n,k,j)$, $0\le j,k\le n$, where $B(n,k,j)$ is the number of type $B$ permutations of order $n$ with $k$ descents and $j$ minus signs. We prove in particular, that $b(n,k,j):=B(n,k,j)/\binom{n}{j}$ is an integer and provide two combinatorial interpretations for these numbers.