An extension of MacMahon's equidistribution theorem to ordered set partitions

An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon’s theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof...

Extension of Preferences to an Ordered Set

If a decision maker prefers x to y to z, would he choose orderd set [x, z] or [y, x]? This article studies extension of preferences over individual alternatives to an ordered set which is prevalent in closed ballot elections with proportional representation and other real life problems where the decision maker is to choose from groups with an associated hierarchy inside. I introduce five ordina...

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

Partitions of a Finite Partially Ordered Set

In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some su...

Extension of Stanley's Theorem for Partitions

In this paper we present an extension of Stanley’s theorem related to partitions of positive integers. Stanley’s theorem states a relation between “the sum of the numbers of distinct members in the partitions of a positive integer n” and “the total number of 1’s that occur in the partitions of n”. Our generalization states a similar relation between “the sum of the numbers of distinct members i...