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

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.