Minimal Permutations and 2-Regular Skew Tableaux
William Y.C. Chen, Cindy C.Y. Gu and Kevin J. Ma
Abstract: 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).
AMS Classification: 05A05, 05A19
Keywords: minimal permutation, 2-regular skew tableau, Knuth equivalence, the RSK algorithm