Let (Q, G) be a permutation representation. The restricted diagonal action of G on the subset of Qn containing n-tuples in which no elements appear more than m times, the so-called multi-restricted power of permutation representation with restriction m, is studied. The orbit decompositions of these actions involve a new class of numbers, the so-called multi-restricted numbers of the first and second kinds with restriction m. The multi-restricted numbers of the second kind without a restriction are the Stirling numbers of the second kind, and the multi-restricted numbers of the second kind with restriction 2 are reparametrized coefficients of Bessel polynomials, the so-called Bessel numbers. The multi-restricted numbers of the second kind with the restriction m ≥ 3 are entirely new, and this thesis is devoted to a study of this new series of numbers.