Unimodal Permutations and Almost-Increasing Cycles

Abstract

In this paper, we establish a natural bijection between the almost-increasing cyclic permutations of length $n$ and unimodal permutations of length $n-1$. This map is used to give a new characterization, in terms of pattern avoidance, of almost-increasing cycles. Additionally, we use this bijection to enumerate several statistics on almost-increasing cycles. Such statistics include descents, inversions, peaks and excedances, as well as the newly defined statistic called low non-inversions. Furthermore, we refine the enumeration of unimodal permutations by descents, inversions and inverse valleys. We conclude this paper with a theorem that characterizes the standard cycle notation of almost-increasing permutations.