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