A context-free grammar for the e-positivity of the trivariate second-order Eulerian polynomials

Ma-Ma-Yeh made a beautiful observation that a change of the grammar of Dumont instantly leads to the γ-positivity of the Eulearian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The appearance of the factor of two fits perfectly in a grammatical labeling of 0-1-2 increasing plane trees. Furthermore, the grammatical calculus is instrumental to the computation of the generating functions. This approach can be adapted to study the e-positivity of the trivariate second-order Eulerian polynomials first introduced by Dumont in the context of ternary trees, and independently defined by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.

[1]  William Y. C. Chen,et al.  Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations , 2012, 1208.1420.

[2]  Amy M. Fu,et al.  Context-free grammars for permutations and increasing trees , 2017, Adv. Appl. Math..

[3]  Yeong-Nan Yeh,et al.  Eulerian Polynomials, Stirling Permutations of the Second Kind and Perfect Matchings , 2017, Electron. J. Comb..

[4]  Miklós Bóna,et al.  Real Zeros and Normal Distribution for Statistics on Stirling Permutations Defined by Gessel and Stanley , 2007, SIAM J. Discret. Math..

[5]  Louis W. Shapiro,et al.  Runs, Slides and Moments , 1983 .

[6]  Yeong-Nan Yeh,et al.  The Ascent-Plateau Statistics on Stirling Permutations , 2019, Electron. J. Comb..

[7]  Amy M. Fu.  A context-free grammar for peaks and double descents of permutations , 2018, Adv. Appl. Math..

[8]  Zhicong Lin,et al.  On γ-positive polynomials arising in Pattern Avoidance , 2017, Adv. Appl. Math..

[9]  Song Y. Yan,et al.  Context Free Grammars , 2011 .

[10]  Dominique Dumont.  Une généralisation trivariée symétrique des nombres eulériens , 1980, J. Comb. Theory, Ser. A.

[11]  Jiang Zeng,et al.  The γ-positivity of basic Eulerian polynomials via group actions , 2015, J. Comb. Theory A.

[12]  D. Dumont.  William Chen grammars and derivations in trees and arborescences. (Grammaires de William Chen et dérivations dans les arbres et arborescences.) , 1996 .

[13]  Svante Janson,et al.  Generalized Stirling permutations, families of increasing trees and urn models , 2008, J. Comb. Theory, Ser. A.

[14]  Dominique Foata,et al.  Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers , 1974 .

[15]  James Haglund,et al.  Stable multivariate Eulerian polynomials and generalized Stirling permutations , 2012, Eur. J. Comb..

[16]  John R. Stembridge,et al.  Eulerian numbers, tableaux, and the Betti numbers of a toric variety , 1992, Discret. Math..

[17]  Svante Janson.  Plane recursive trees, Stirling permutations and an urn model , 2008 .

[18]  William Y. C. Chen,et al.  Context-Free Grammars, Differential Operators and Formal Power Series , 1993, Theor. Comput. Sci..

[19]  L. Lesieur,et al.  On the eulerian numbers , 1992 .

[20]  Christos A. Athanasiadis.  Gamma-positivity in combinatorics and geometry , 2017, 1711.05983.

[21]  D. Foata,et al.  Theorie Geometrique des Polynomes Euleriens , 1970 .

[22]  Sergi Elizalde.  Descents on quasi-Stirling permutations , 2021, J. Comb. Theory, Ser. A.

[23]  Petter Brand'en,et al.  Actions on permutations and unimodality of descent polynomials , 2006, math/0610185.

[24]  Chak-On Chow.  On certain combinatorial expansions of the Eulerian polynomials , 2008, Adv. Appl. Math..

[25]  Heesung Shin,et al.  The symmetric and unimodal expansion of Eulerian polynomials via continued fractions , 2010, Eur. J. Comb..

[26]  Vesselin Gasharov,et al.  On the Neggers-Stanley Conjecture and the Eulerian Polynomials , 1998, J. Comb. Theory, Ser. A.

[27]  Yeong-Nan Yeh,et al.  Γ-positivity and Partial Γ-positivity of Descent-type Polynomials , 2019, J. Comb. Theory, Ser. A.

[28]  Hao Zhou,et al.  A New Encoding of Permutations by Laguerre Histories , 2019, Electron. J. Comb..