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<p>Carlitz and Scoville (1974) introduced the polynomials <i>A</i><sub><i>n</i></sub>(<i>x, y</i>∣<i>α, β</i>), which we refer to as the (<i>α, β</i>)-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 (1974) obtained the generating function for <i>A</i><sub><i>n</i></sub>(<i>x, y</i>∣<i>α, β</i>). In this paper, we introduce a new family of polynomials, <i>P</i><sub><i>n</i></sub>(<i>u, v, w, z</i>∣<i>α, β</i>), defined on descent-Stirling statistics of permutations 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 (1993), we establish a connection between the generating function for <i>P</i><sub><i>n</i></sub>(<i>u, v, w, z</i>∣<i>α, β</i>) and the generating function for <i>A</i><sub><i>n</i></sub>(<i>x, y</i>∣<i>α, β</i>). Using this connection, we derive the generating function for <i>P</i><sub><i>n</i></sub>(<i>u, v, w, z</i>∣<i>α, β</i>), which can be specialized to obtain (<i>α, β</i>)-extensions of generating functions for peaks, left peaks, double ascents, right double ascents and left-right double ascents given by David and Barton (1962), Elizalde and Noy (2003), Entringer (1969), Gessel and Zhuang (2018), Kitaev (2007), and Zhuang (2016). Moreover, we establish two relations between <i>P</i><sub><i>n</i></sub>(<i>u, v, w, z</i>∣<i>α, β</i>) and <i>A</i><sub><i>n</i></sub>(<i>x, y</i>∣<i>α, β</i>), which enable us to derive (<i>α, β</i>)-extensions of results obtained by Stembridge (1997), Petersen (2006), Brändén (2008) and Zhuang (2017), respectively. We also establish the left peak version of Stembridge’s formula and the peak version of Petersen’s formula, along with their respective (<i>α, β</i>)-extensions, by utilizing these two relations. Specializing (<i>α, β</i>)-extensions of Stembridge’s formula and the left peak version of Stembridge’s formula allows us to derive (<i>α, β</i>)-extensions of the tangent and secant numbers.</p>

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The (α, β)-Eulerian polynomials and descent-Stirling statistics on permutations

  • 期刊论文

  • 发表日期：

    2025年6月26日

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  • Kathy Q. Ji

摘要

Carlitz and Scoville (1974) 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 (1974) obtained the generating function for An(x, y∣α, β). In this paper, we introduce a new family of polynomials, Pn(u, v, w, z∣α, β), defined on descent-Stirling statistics of permutations 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 (1993), we establish a connection between the generating function for Pn(u, v, w, z∣α, β) and the generating function for An(x, y∣α, β). Using this connection, we derive the generating function for Pn(u, v, w, z∣α, β), which can be specialized to obtain (α, β)-extensions of generating functions for peaks, left peaks, double ascents, right double ascents and left-right double ascents given by David and Barton (1962), Elizalde and Noy (2003), Entringer (1969), Gessel and Zhuang (2018), Kitaev (2007), and Zhuang (2016). Moreover, we establish two relations between Pn(u, v, w, z∣α, β) and An(x, y∣α, β), which enable us to derive (α, β)-extensions of results obtained by Stembridge (1997), Petersen (2006), Brändén (2008) and Zhuang (2017), respectively. We also establish the left peak version of Stembridge’s formula and the peak version of Petersen’s formula, along with their respective (α, β)-extensions, by utilizing these two relations. Specializing (α, β)-extensions of Stembridge’s formula and the left peak version of Stembridge’s formula allows us to derive (α, β)-extensions of the tangent and secant numbers.