Counting permutations by the number of successions within cycles

In this note, we consider the problem of counting (cycle) successions  , i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on SnSn for the statistics recording the number of cycles and adjacencies of the form j,j+dj,j+d where d>0d>0 which extends earlier results.