Permutations with Interval Cycles

Abstract

We study permutations of the set \([n] = \{1, 2, \ldots, n\}\) written in cycle notation, for which each cycle forms an increasing or decreasing interval of positive integers. More generally, permutations whose cycle elements form arithmetic progressions are considered. We also investigate the class of generalized interval permutations, where each cycle can be rearranged in increasing order to form an interval of consecutive positive integers.