Jiang Zeng

We provide a bijective proof of the equidistribution of two pairs of vincular patterns in permutations, thereby resolving a recent open problem of Bitonti, Deb, and Sokal (arXiv:2412.10214). Since the bijection is involutive, we also confirm their conjecture on the equidistribution of triple vincular patterns. Somewhat unexpectedly, we show that this involution is closed on the set of Baxter permutations, thereby implying another trivariate symmetries of vincular patterns. The proof of this second result requires a variant of a characterization of Baxter permutations in terms of restricted Laguerre histories, first given by Viennot using the Françon-Viennot bijection.

Motivated by recent work on (re)mixed Eulerian numbers, we provide a combinatorial interpretation of a subfamily of the remixed Eulerian numbers introduced by Nadeau and Tewari. More specifically, we show that these numbers can be realized as the generating polynomials of permutations with respect to the statistics of left-to-right minima, right-to-left minima, descents, and the mixed major index. Our results generalize both the bi-Stirling-Eulerian polynomials of Carlitz-Scoville and the Stirling-Euler-Mahonian polynomials of Butler.

We explore some connections between moments of rescaled little $q$-Jacobi polynomials, $q$-analogues of values at negative integers for some Dirichlet series, and the $q$-Eulerian polynomials of wreath products of symmetric groups.

Recently Cheng et al. (Adv. in Appl. Math. 143 (2023) 102451) generalized the inversion number to partial permutations, which are also known as Laguerre digraphs, and asked for a suitable analogue of MacMahon's major index. We provide such a major index, namely, the corresponding maj and inv statistics are equidistributed, and exhibit a Haglund-Remmel-Wilson type identity. We then interpret some Jacobi-Rogers polynomials in terms of Laguerre digraphs generalizing Deb and Sokal's alternating Laguerre digraph interpretation of some special Jacobi-Rogers polynomials.