Enumeration formulas for generalized q-Euler numbers

We find an enumeration formula for a (t,q)-Euler number which is a generalization of the q-Euler number introduced by Han, Randrianarivony, and Zeng. We also give a combinatorial expression for the (t,q)-Euler number and find another formula when t=±qr for any integer r. Special cases of our latter formula include the formula of the q-Euler number recently found by Josuat-Vergès and Touchard-Riordanʼs formula.