Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Context-free Grammar for the $e$-Positivity of the Trivariate Second-order Eulerian Polynomials
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Ma-Ma-Yeh made a beautiful observation that a change of the grammar of Dumont instantly leads to the $gamma$-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 introduced by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.
rate research
Read More
Inspired by the recent work of Chen and Fu on the e-positivity of trivariate second-order Eulerian polynomials, we show the e-positivity of a family of multivariate k-th order Eulerian polynomials. A relationship between the coefficients of this e-positive expansion and second-order Eulerian numbers is established. Moreover, we present a grammatical proof of the fact that the joint distribution of the ascent, descent and j-plateau statistics over k-Stirling permutations are symmetric distribution. By using symmetric transformation of grammars, a symmetric expansion of trivariate Schett polynomial is also established.
Ramanujan defined the polynomials $psi_{k}(r,x)$ in his study of power series inversion. Berndt, Evans and Wilson obtained a recurrence relation for $psi_{k}(r,x)$. In a different context, Shor introduced the polynomials $Q(i,j,k)$ related to improper edges of a rooted tree, leading to a refinement of Cayleys formula. He also proved a recurrence relation and raised the question of finding a combinatorial proof. Zeng realized that the polynomials of Ramanujan coincide with the polynomials of Shor, and that the recurrence relation of Shor coincides with the recurrence relation of Berndt, Evans and Wilson. So we call these polynomials the Ramanujan-Shor polynomials, and call the recurrence relation the Berndt-Evans-Wilson-Shor recursion. A combinatorial proof of this recursion was obtained by Chen and Guo, and a simpler proof was recently given by Guo. From another perspective, Dumont and Ramamonjisoa found a context-free grammar $G$ to generate the number of rooted trees on $n$ vertices with $k$ improper edges. Based on the grammar $G$, we find a grammar $H$ for the Ramanujan-Shor polynomials. This leads to a formal calculus for the Ramanujan-Shor polynomials. In particular, we obtain a grammatical derivation of the Berndt-Evans-Wilson-Shor recursion. We also provide a grammatical approach to the Abel identities and a grammatical explanation of the Lacasse identity.
This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics. Properties of the enumerative polynomials for permutations, signed permutations and derangements, including generating functions and gamma-positivity are studied, which generalize and unify earlier results of Athanasiadis, Brenti, Chow, Petersen, Roselle, Stembridge, Shin and Zeng. In particular, we derive a formula expressing the joint distribution of excedance number and negative number statistics over the type B derangements in terms of the derangement polynomials.
We provide combinatorial interpretation for the $gamma$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $gamma$-coefficients for derangements. Our results refine the classical $gamma$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Brandens modified Foata--Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).
In this paper, we give a type B analogue of the 1/k-Eulerian polynomials. Properties of this kind of polynomials, including combinatorial interpretations, recurrence relations and gamma-positivity are studied. In particular, we show that the 1/k-Eulerian polynomials of type B are gamma-positive when $k>0$. Moreover, we obtain the corresponding results for derangements of type B. We show that a type B 1/k-derangement polynomials $d_n^B(x;k)$ are bi-gamma-positive when $kgeq 1/2$. In particular, we get a symmetric decomposition of $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
