Minimal permutations and 2-regular skew tableaux

Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let Fd(n) denote the set of minimal permutations of length n with d descents, and let fd(n)=|Fd(n)|. They showed that fn−2(n)=2n−(n−1)n−2 and fn(2n)=Cn, where Cn is the n-th Catalan number. Mansour and Yan proved that fn+1(2n+1)=2n−2nCn+1. In this paper, we consider the problem of counting minimal permutations in Fd(n) with a prescribed set of ascents, and we show that they are in one-to-one correspondence with a class of skew Young tableaux, which we call 2-regular skew tableaux. Using the determinantal formula for the number of skew Young tableaux of a given shape, we find an explicit formula for fn−3(n). Furthermore, by using the Knuth equivalence, we give a combinatorial interpretation of a formula for a refinement of the number fn+1(2n+1).