I was wondering whether there is a standard name in the literature for the following family $\mathcal{F}$ of languages over any finite alphabet $\Sigma = \{a_1,\ldots,a_k\}$:
$\mathcal{F}$ consists of all languages of the shape $\{a_{i_1}^{c_1}a_{i_2}^{n_2}a_{i_3}^{c_3}\ldots a_{i_{k-1}}^{n_{k-1}}a_{i_k}^{c_k}: n_2,n_4,\ldots,n_{k-1} \geq 0 \wedge R(n_2,n_4,\ldots,n_{k-1})\}$, where $c_1,c_3,\ldots,c_k$ are fixed nonnegative constants and $R$ is any fixed relation on $m$ variables $u_1,u_2,\ldots,u_m$ such that for some $S \subseteq \{1,2,\ldots,m\} \times \{1,2,\ldots,m\}$, $R(u_1,u_2,\ldots,u_m)$ holds iff $\left((\forall i,j \in \{1,\ldots,m\})[u_i = u_j \Leftrightarrow (i,j) \in S]\right)$.
So $\mathcal{F}$ contains languages like $\{a_1^n a_2^n a_3^n: n \geq 0\}$ (the "standard" context-sensitive but non-context-free language) and $\{a_1^2 a_1^n a_2^m a_1 a_1^m a_2^n a_1: n,m \geq 0\}$.
This family of languages cropped up in my work recently and I was wondering whether anything interesting could be said about it..
Thank you!
