Let $N = [n]$ and for any subset $A \subseteq N$, let $S_A$ denote the subgroup of the symmetric group $S_n$ that fixes all objects outside $A$. Say that a sequence $A_1, \dots, A_k \subseteq N$ is "identity free" if the equivalence $$s_1 s_2 \dots s_k = \mathop{id} \qquad \text{where } s_i \in S_{A_i} \text{ for all } i$$ has no solutions, except for the trivial one where $s_1 = s_2 = \dots = s_k = \mathop{id}$.

Have these been studied under any name that I can search for? While I would be interested in any discussion at all, I am particularly interested in the question of the extremal length $L$ of the longest identity-free sequence when all $A_i$ are constrained to have some fixed size $\alpha$. The only upper bounds on $L$ I have been able to observe so far come from the easy observation that no two $A_i$ may overlap on more than one element (and then apply Cauchy-Schwarz).