The expected shape of random doubly alternating Baxter permutations

Source abstract

Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e., Baxter permutations $\sigma \in S_n$, such that $\sigma$ and $\sigma^{-1}$ are alternating. They proved that the number of such permutations in $S_{2n}$ and $S_{2n+1}$ is the Catalan number $C_n$. In this paper, we compute the expected limit shape of such permutations, following the approach by Miner and Pak.