Application of an Inverse Dirichlet’s Principle to Discrete Recreational Problems: Bound Estimation’s Optimization Using Combinatorial Probability and Comparison of Numerical Bound Estimation Using Various Algorithms, Including Recursive Inclusion-Exclusion Principle

Dirichlet’s principle, also known as a pigeonhole principle, claims that if \(n \in \mathbb {N}\) items are put into \(m \in \mathbb {N}\) 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 \(n \in \mathbb {N}\) items put in \(m \in \mathbb {N}\) containers, when \(n < m\) , then there is at least one container with no item inside. Moreover, we use discrete combinatorics to refine Dirichlet’s principle within a probabilistic framework. Applying stochastic fashion on Dirichlet’s 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 principle may have some practical applications rather than the original principle, particularly considering derivation and estimate of a probability there is still one of m containers with no item in it, even if there are no fewer items than containers. Also, we derive an effective lower bound for the probability of an empty container given the problem parameters, as demonstrated using some applied discrete recreational problems, particularly an unoccupied doubleseat problem and an unoccupied l-seat problem. Finally, since the discrete recreational problems contain some exhaustive computational parts, especially regarding the probability’s calculation, we compare various approaches to the numerical estimation of the probability’s numerator, searching for an optimal approach conditional on the problem’s parameter setting.