Desarrangements revisited: statistics and pattern avoidance

A desarrangement is a permutation whose first ascent is even. Desarrangements were introduced in the 1980s by Jacques Désarménien, who proved that they are in bijection with derangements. We revisit the study of desarrangements, focusing on two themes: the refined enumeration of desarrangements with respect to permutation statistics, and pattern avoidance in desarrangements. Our main results include generating function formulas for counting desarrangements by the number of descents, peaks, valleys, double ascents, and double descents, as well as a complete enumeration of desarrangements avoiding a prescribed set of length 3 patterns. We find new interpretations of the Catalan, Fine, Jacobsthal, and Fibonacci numbers in terms of pattern-avoiding desarrangements.