The (α,β)-Eulerian Polynomials and Descent-Stirling Statistics on Permutations
Abstract
Carlitz and Scoville introduced the polynomials An(x,y|α,β), which we refer to as the (α, β)-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 An(x,y|α,β). In this paper, we introduce a new family of polynomials, Pn(u1,u2,u3,u4|α,β), 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 Pn(u1,u2,u3,u4|α,β) and the generating function of the (α,β)-Eulerian polynomials An(x,y|α,β) introduced by Carlitz and Scoville. Using this connection, we derive the generating function of Pn(u1,u2,u3,u4|α,β), which can be specialized to obtain the (α,β)-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 Pn(u1,u2,u3,u4|α,β) and An(x,y|α,β), which enable us to derive (α,β)-extensions of results of Stembridge, Petersen, Br\"and\'en, and Zhuang. Specializing (α,β)-extensions of Stembridge's formula and the left peak version of Stembridge's formula allows us to derive the (α,β)-extensions of the tangent and secant numbers.
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