Tri-Restricted Numbers and Powers of Permutation Representations

Let \(G\) be a transitive permutation group on a set \(Q\). The orbit decompositions of the actions of \(G\) on the sets of ordered \(n\)-tuples with elements repeated at most three times are studied. The decompositions involve Stirling numbers and a new class of related numbers, the so-called tri-restricted numbers. The paper presents exponential generating functions for the numbers of orbits, and examines relationships between various powers of the \(G\)-set involving Stirling numbers, the tri-restricted numbers, and the coefficients of Bessel polynomials.