Combinatorial parameters on bargraphs of permutations

‎In this paper‎, ‎we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps‎. ‎More precisely‎, ‎we find the joint distribution of the descent and up step statistics on the bargraph representations‎, ‎thereby obtaining a new refined count of permutations of a given length‎. ‎To do so‎, ‎we consider the distribution of the parameters on permutations of a more general multiset of which $mathcal{S}_n$ is a subset‎. ‎In addition to finding an explicit formula for the joint distribution on this multiset‎, ‎we provide counts for the total number of descents and up steps of all its members‎, ‎supplying both algebraic and combinatorial proofs‎. ‎Finally‎, ‎we derive explicit expressions for the sign balance of these statistics‎, ‎from which the comparable results on permutations follow as special cases‎.