An involution for a Catalan-tangent number identity

We provide an involution proof of a Catalan-tangent number identity arising from the study of peak algebra that was found by Aliniaeifard and Li. In the course, we find a new combinatorial identity for the tangent numbers $T_{2n+1}$: $$\sum _{k=0}^n(-1{)}^k\left(\genfrac{}{}{0px}{}{2n+1}{2k}\right)2^{2n-2k}{T}_{2k+1}=(-1{)}^n{T}_{2n+1}\mathrm{.}$$ Moreover, we derive two different $q$-analogs of the above identity from the combinatorial perspective.