In this paper, we study max-weight stochastic matchings on online bipartite graphs under both vertex and edge arrivals. We focus on designing polynomial time approximation algorithms with respect to the online benchmark, which was first considered by Papadimitriou, Pollner, Saberi, and Wajc [EC'21].

The permutation language $P_n$ consists of all words that are permutations of a fixed alphabet of size $n$. Using divide-and-conquer, we construct a regular expression $R_n$ that specifies $P_n$. We then give explicit bounds for the length of $R_n$, which we find to be $4^n n^{-(\lg n)/4+Θ(1)}$, and use these bounds to show that $R_n$ has minimum size over all regular expressions specifying $P_n$.