The $(\alpha,\beta)$-Eulerian Polynomials and Descent-Stirling Statistics on Permutations

Carlitz and Scoville introduced the polynomials $A_n(x,y|{\alpha},{\beta})$, which we refer to as the $(\alpha, \beta)$-Eulerian polynomials. These polynomials count permutations based on Eulerian-Stirling statistics, including descents, ascents, left-to-right maxima, and right-to-left maxima. Carlitz and Scoville obtained the generating function of $A_n(x,y|{\alpha},{\beta})$. In this paper, we introduce a new family of polynomials, $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$, defined on permutations, incorporating descent-Stirling statistics including valleys, exterior peaks, right double descents, left double ascents, left-to-right maxima, and right-to-left maxima. By employing the grammatical calculus introduced by Chen, we establish the connection between the generating function of $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$ and the generating function of the $(\alpha,\beta)$-Eulerian polynomials $A_n(x,y|{\alpha},{\beta})$ introduced by Carlitz and Scoville. Using this connection, we derive the generating function of $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$, which can be specialized to obtain the $(\alpha,\beta)$-extensions of generating functions for peaks, left peaks, double ascents, right double ascents and left-right double ascents given by David-Barton, Elizalde and Noy, Entringer, Gessel, Kitaev and Zhuang. Moreover, we establish two relations between $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$ and $A_n(x,y|{\alpha},{\beta})$, which enable us to derive $(\alpha,\beta)$-extensions of results of Stembridge, Petersen, Br\"and\'en, and Zhuang. Specializing $(\alpha,\beta)$-extensions of Stembridge's formula and the left peak version of Stembridge's formula allows us to derive the $(\alpha,\beta)$-extensions of the tangent and secant numbers.