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A matrix T=(t_ik) is introduced, the coefficients of which are defined by $\text{k}{}_{\mathrm{ik}}\text{:= (}\text{i}{}^{\mathrm{k}}\text{(ik)!)}\text{Σ}{}_{\mathrm{x?S}}{}_{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{(x)}{}^{\mathrm{k}}\text{, i, k?N={1, 2, 3,…,}}$, where a_i(x) denotes the s the number of i cycles in the element x of the symmetric group S_n. It is shown that these numbers are natural numbers, that they are easy to evaluate, and that they serve very well in order to formulate an infinite number of characterizations of multiply transitive subgroups of symmetric groups in terms of the cycle structure of their elements.