Combinatorial Theory

In 1977, Carlitz and Scoville introduced the cycle \((\alpha,t)\)-Eulerian polynomials \(A^{\mathrm{cyc}}_n(x,y,t \mid \alpha)\) by enumerating permutations with respect to the number of excedances, drops, fixed points, and cycles. In this paper, we propose a nine-variable generalization of the Eulerian polynomials \(A_n(u_1,u_2,u_3,u_4,f,g,t \mid \alpha,\beta)\) defined in terms of descent-based statistics of permutations, and we establish a connection formula linking these two families of generalized Eulerian polynomials. By means of this connection formula, we derive explicit exponential generating functions and several \(\gamma\)-positive expressions for various Eulerian-type polynomials. In particular, our results unify and strengthen recent findings of Ji and Ji-Lin. In related work on the transition matrix between the Specht and web bases, Hwang, Jang, and Oh recently introduced the web permutations, which can be described in terms of cycle André permutations. We show that enumerating these permutations with respect to the numbers of drops, fixed points, and cycles yields the normalized \(\gamma\)-vectors of the \((\alpha,t)\)-Eulerian polynomials. Our results thus provide a unified framework encompassing and extending several known results in the literature.

Mathematics Subject Classifications: 05A05, 05A15, 05A19

Keywords: \((\alpha,t)\)-Eulerian polynomials, left-to-right maxima, cycle André permutations, cyclic valley-hopping, gamma-positivity