Alternating subsets modulo m

We enumerate increasing combinations of {1, 2, . . . , n} according to parity statistics defined on pairs of adjacent elements. Generating functions are used to devise a framework that addresses all questions on adjacencies of parities with respect to any modulus m > 1. In particular, we give a generalization of a classical result on alternating subsets which was previously known for the modulus 2. We also compute some generating functions for the number of combinations possessing special adjacent parity patterns.