Revisiting $q$-Derangement Numbers via Decorated Permutations

Abstract

This paper aims to provide a direct combinatorial proof of the Gessel-Wachs formula for $q$-derangement numbers in the setting of decorated permutations, without using the $q$-binomial inversion formula. Decorated permutations, introduced by Postnikov in his study of the totally nonnegative Grassmannian, provide a natural framework for Chen's signed fixed-point model. Our proof is based on a major-index generating function for decorated permutations with a fixed number of signed fixed points, together with a sign-reversing and descent-set-preserving involution, thereby answering a question raised by Chen. This involution was discovered through human--AI collaboration. Moreover, our framework yields an immediate proof of a result of Désarménien and Wachs concerning the equidistribution between descent classes of derangements and descent classes of desarrangements. We also supply combinatorial proofs of two recurrence relations for the $q$-derangement numbers in the setting of desarrangements with the aid of the insertion lemma for ordinary permutations.