Dirichlet's principle revisited: An inverse Dirichlet's principle definition and its bound estimation improvement using stochastic combinatorics

Abstract

Dirichlet's principle, also known as a pigeonhole principle, claims that if n ∈ N item are put into m ∈ N containers, with n > m, then there is a container that contains more than one item. In this work, we focus rather on an inverse Dirichlet's principle (by switching items and containers), which is as follows: considering n ∈ N items put in m ∈ N containers, when n < m, then there is at least one container with no item inside. Furthermore, we refine Dirichlet's principle using discrete combinatorics within a probabilistic framework. Applying stochastic fashion on the principle, we derive the number of items n may be even greater than or equal to m, still very likely having one container without an item. The inverse definition of the problem rather than the original one may have some practical applications, particularly considering derived effective upper bound estimates for the items number, as demonstrated using some applied mini-studies.