On the enumeration of d-minimal permutations

TL;DR

This paper introduces a new method using skew Young tableaux to count minimal permutations with a given number of descents, leading to general formulas and new insights into Eulerian numbers.

Contribution

It presents a novel approach for enumerating d-minimal permutations via skew Young tableaux and derives general determinant-based formulas, extending understanding of Eulerian numbers.

Findings

Derived a general expression for counting minimal permutations with d descents.

Extended the class of skew Young tableaux to uncover new results on Eulerian numbers.

Provided determinant sum formulas for enumeration.

Abstract

We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.