Relations of multiple $\tilde{T}$-values involving the total numbers of certain permutations

TL;DR

This paper generalizes known relations involving multiple $ ilde{T}$-values and Entringer numbers, connecting permutation counts with explicit formulas for Dumont permutations of the first kind.

Contribution

It extends previous work by deriving new relations and explicit formulas linking multiple $ ilde{T}$-values, Entringer numbers, and Dumont permutations of the first kind.

Findings

Derived new relations involving multiple $ ilde{T}$-values and Entringer numbers.

Provided explicit formulas for counting Dumont permutations of the first kind.

Extended the theoretical framework connecting permutation enumeration and special values.

Abstract

Kaneko and Tsumura proved a relation of multiple $\tilde{T}$-values involving Entringer numbers counting the total number of down-up permutations starting with a fixed value. In the present paper, we generalize this relation and provide some relations involving Entringer numbers and the total number of Dumont permutations of the first kind starting with a fixed value. For this purpose, we also provide explicit formulas for the total numbers of those permutations.