Average length of the longest increasing subsequences in random involutions avoiding 231 and a layered pattern

The problems of deciding the length of the longest increasing subsequence for a permutation or an involution, the expecta-tion and the limiting distribution, surveyed in Stanley’s ICM (International Congress of Mathematicians) Plenary address, have been widely studied by combinatorialists, analysts and probabilists. Partially motivated by the intriguing phenomenon stated by Simion and Schmidt [European J. Combin. 6 (1985) 383–406] that 231-avoiding permutations are exactly the set of layered permutations, in this paper, we investigate the limiting behavior of the average length of the longest increasing subsequences in random involutions avoiding 231 and another pattern which is a layered permutation. We obtain an ex-plicit formula of the generating function and apply it to exemplify a set of interesting examples, which extend recent results of the first author with Yıldırım [Turkish J. Math. 43 (2019) 2183–2192], where the longest increasing subsequences in involutions avoiding a pair of patterns of length 3 are studied.