I am trying to figure out a good way to denote a particular kind of permutation of the set of pairs $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$. I want permutations that consist of permuting the pairs (so essentially a permutation of the set $\{1, 2, \ldots, n\}$), and then individually permuting the pairs themselves afterward. For example, if $n=4$, one such permutation would be $$(b_3, a_3), (a_1, b_1), (b_2, a_2), (a_4, b_4).$$
I was thinking about doing the following:
Consider a permutation $\sigma_{\tau_1, \ldots, \tau_n}$ on the set $\{(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)\}$ as a sequence $$\sigma_{\tau_1, \ldots, \tau_n}=(\sigma_{\tau_1}((a_1, b_1)), \ldots, \sigma_{\tau_n}((a_n, b_n))),$$ where $\sigma$ is a permutation of $\{1, 2, \ldots, n\}$ and $\tau_i$ is a permutation of $\{a_i, b_i\}$, for each $1\leq i \leq n$.
I am not a huge fan of this, though. Any suggestions? I greatly appreciate them.