Minimal Permutations and 2-Regular Skew Tableaux

TL;DR

This paper studies minimal permutations with a fixed number of descents, establishing connections to 2-regular skew Young tableaux and deriving explicit formulas for their enumeration.

Contribution

It introduces 2-regular skew tableaux to count minimal permutations with prescribed ascents and provides explicit formulas for specific cases.

Findings

Derived an explicit formula for f_{n-3}(n).

Established a bijection between minimal permutations and 2-regular skew tableaux.

Provided a combinatorial interpretation of a refined enumeration formula.

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 $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that such structures 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 $f_{n-3}(n)$. Furthermore, by using the Knuth equivalence, we give a combinatorial interpretation of a formula for a refinement of the number $f_{n+1}(2n+1)$.