Enumeration of inversion sequences according to the outer and inner perimeter Article Swipe
YOU?
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· 2024
· Open Access
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· DOI: https://doi.org/10.2478/puma-2023-0004
· OA: W4406491012
The integer sequence π = π 1 ‧‧‧ 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.
