Combinatorial parameters on bargraphs of permutations

In this paperý, ýwe consider statistics on permutations of length n represented geometrically as bargraphs having the same number of horizontal stepsý. ýMore preciselyý, ýwe find the joint distribution of the descent and up step statistics on the bargraph representationsý, ýthereby obtaining a new refined count of permutations of a given lengthý. ýTo do soý, ýwe consider the distribution of the parameters on permutations of a more general multiset of which Sn is a subsetý. ýIn addition to finding an explicit formula for the joint distribution on this multisetý, ýwe provide counts for the total number of descents and up steps of all its membersý, ýsupplying both algebraic and combinatorial proofsý. ýFinallyý, ýwe derive explicit expressions for the sign balance of these statisticsý, ýfrom which the comparable results on permutations follow as special casesý.