The signed Eulerian numbers on involutions

TL;DR

This paper introduces a new sequence called signed Eulerian numbers for involutions in the symmetric group, providing explicit formulas and recurrences based on their generating functions.

Contribution

It defines the signed Eulerian numbers for involutions and derives their combinatorial properties, including explicit formulas and recurrence relations.

Findings

Derived explicit formula for signed Eulerian numbers

Established recurrence relations for the sequence

Analyzed combinatorial properties of involutions

Abstract

We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for $f_{n,k}$ arising from the properties of its generating function.