Desarrangements revisited: statistics and pattern avoidance
Chadi Bsila, Caroline E. Cox, Anna S. Hugo, Lindsey A. Styron, Yan Zhuang
TL;DR
The paper revisits desarrangements, examining two core themes: refined enumeration by permutation statistics and pattern avoidance within desarrangements. It develops exponential generating functions for desarrangements under several statistics (des, pk, val, dasc, ddes) and introduces a generalized run theorem to systematically derive these counts and joint distributions. It then completes the pattern-avoidance landscape for all length-3 pattern sets, linking several avoidance classes to Catalan, Fine, Jacobsthal, and Fibonacci numbers, and providing bijective and generating-function proofs. Finally, it places pattern-avoidance in desarrangements in context with derangements, characterizing Wilf-equivalences and open questions about equidistribution of related statistics. Overall, the work deepens the combinatorial understanding of desarrangements through exact enumerations, generating functions, and structural insights tied to classical sequences and pattern-avoidance phenomena.
Abstract
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.