The Expected Shape of Random Doubly Alternating Baxter Permutations

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 explore the expected limit shape of such permutations, following the approach by Miner and Pak.