Enumeration of inversion sequences according to the outer and inner perimeter

The integer sequence π = π₁ ‧‧‧ π_n is said to be an inversion sequence if 0 ≤ π_i ≤ i – 1 for all i. Let ℐ_n denote the set of inversion sequences of length n, represented using positive instead of non-negative integers. We consider here two new statistics defined on the bargraph representation b(π) of an inversion sequence π which record the number of unit squares touching the boundary of b(π) and that are either exterior or interior to b(π). We denote these statistics on ℐ_n recording the number of outer and inner perimeter squares respectively by oper and iper. In this paper, we study the distribution of oper and iper on ℐ_n and also on members of ℐ_n that end in a particular letter. We find explicit formulas for the maximum and minimum values of oper and iper achieved by a member of ℐ_n as well as for the average value of these parameters. We make use of both algebraic and combinatorial arguments in establishing our results.