scholarly journals A Note on the $\gamma$-Coefficients of the Tree Eulerian Polynomial

10.37236/5361 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Rafael S. González D'León
Keyword(s):  
Lie Algebra ◽  
Product Formula ◽  
Rooted Trees ◽  
Binary Trees ◽  
Eulerian Polynomial ◽  
Generating Polynomial ◽  
Combinatorial Interpretations ◽  
Set Of Permutations ◽  
Positive Coefficients

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D'León-Wachs and Gonzláez D'León related to the free multibracketed Lie algebra and the poset of weighted partitions.

scholarly journals A Combinatorial Proof of Postnikov's Identity and a Generalized Enumeration of Labeled Trees

10.37236/1890 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Seunghyun Seo
Keyword(s):  
Rooted Trees ◽  
Binary Trees ◽  
Combinatorial Proof ◽  
Labeled Trees ◽  
Plane Trees

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled $k$-ary trees, rooted labeled trees, and labeled plane trees.

scholarly journals A Refinement of Cayley's Formula for Trees

10.37236/1884 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
Ira M. Gessel ◽  
Seunghyun Seo
Keyword(s):  
Rooted Tree ◽  
Rooted Trees ◽  
Binary Trees ◽  
Parking Functions ◽  
Hook Length ◽  
Proper Vertex

A proper vertex of a rooted tree with totally ordered vertices is a vertex that is the smallest of all its descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials $$P_n(a,b,c)= c\prod_{i=1}^{n-1}(ia+(n-i)b +c),$$ which reduce to $(n+1)^{n-1}$ for $a=b=c=1$. Our study of proper vertices was motivated by Postnikov's hook length formula $$(n+1)^{n-1}={n!\over 2^n}\sum _T \prod_{v}\left(1+{1\over h(v)}\right),$$ where the sum is over all unlabeled binary trees $T$ on $n$ vertices, the product is over all vertices $v$ of $T$, and $h(v)$ is the number of descendants of $v$ (including $v$). Our results give analogues of Postnikov's formula for other types of trees, and we also find an interpretation of the polynomials $P_n(a,b,c)$ in terms of parking functions.

scholarly journals On the Lie enveloping algebra of a pre-Lie algebra

2008 ◽  
Vol 2 (1) ◽  
pp. 147-167 ◽  
Author(s):  
J.-M. Oudom ◽  
D. Guin
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebra ◽  
Associative Product ◽  
Planar Rooted Trees ◽  
Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

scholarly journals W-algebras and Duflo isomorphism

2018 ◽  
Vol 17 (03) ◽  
pp. 1850041
Author(s):  
P. Batakidis ◽  
N. Papalexiou
Keyword(s):  
Lie Algebra ◽  
Deformation Quantization ◽  
Poisson Manifold ◽  
Product Formula ◽  
Poisson Structures ◽  
Semisimple Lie Algebra

We prove that when Kontsevich’s deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra [Formula: see text] the Poisson manifold [Formula: see text] is [Formula: see text]. As an application we provide an isomorphism between the Cattaneo–Felder–Torossian reduction algebra [Formula: see text] and the [Formula: see text]-algebra [Formula: see text]. We also show that in the [Formula: see text]-algebra setting, [Formula: see text] is polynomial. Finally, we compute generators of [Formula: see text] as a deformation of [Formula: see text].

scholarly journals On bijections between monotone rooted trees and the comb basis

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Fu Liu
Keyword(s):  
Lie Algebra ◽  
Rooted Trees ◽  
Close Connection ◽  
Free Lie Algebra ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Lie Brackets ◽  
Element Set

International audience Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$. Soit $A$ un ensemble à $n$ éléments. Soit $\mathscr{L} ie_2(A)$ la partie multilinéaire de l'algèbre de Lie libre sur $A$ avec une paire de crochets de Lie compatibles et $\mathscr{L} ie_2(A, i)$ le sous-espace de$\mathscr{L} ie_2(A)$ généré par tous les monômes en $\mathscr{L} ie_2(A)$ avec $i$ supports d'un même type. L'auteur et Dotsenko-Khoroshkin montrent que la dimension de $\mathscr{L} ie_2(A, i)$ est la taille de la $R_{A,i}$, l'ensemble des arbres enracinés sur $A$ avec $i$ arêtes décroissantes. Il y a trois familles de bases connues pour $\mathscr{L} ie_2(A, i)$ : la base de peigne, la base Lyndon, et la base Liu-Lyndon. Récemment, Gonzalez, D' Léon et Wachs, dans leur étude de (co)-homologie de la poset des partitions pondérés, ont demandé si il y a des bijections jolies entre$R_{A,i}$, et la base de peigne ou la base Lyndon. Nous donnons une définition naturelle de "bijection jolie " , et un conjecture qu'il y a une seule bijection jolie entre $R_{A,i}$, et la base de peigne. Nous montrons que la conjecture est vraie pour les cas extrêmes: $i = 0$, et $n − 1$.

scholarly journals Common Edges in Rooted Trees and Polygonal Triangulations

10.37236/2541 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Sean Cleary ◽  
Andrew Rechnitzer ◽  
Thomas Wong
Keyword(s):  
Polynomial Time ◽  
Rooted Trees ◽  
Binary Trees ◽  
Regular Polygons ◽  
Large Component ◽  
Polynomial Time Algorithms ◽  
The Common ◽  
Flip Distance ◽  
Degree Of Similarity ◽  
Insight Into

Rotation distance between rooted binary trees measures the degree of similarity of two trees with ordered leaves and is equivalent to edge-flip distance between triangular subdivisions of regular polygons. There are no known polynomial-time algorithms for computing rotation distance. Existence of common edges makes computing rotation distance more manageable by breaking the problem into smaller subproblems. Here we describe the distribution of common edges between randomly-selected triangulations and measure the sizes of the remaining pieces into which the common edges separate the polygons. We find that asymptotically there is a large component remaining after sectioning off numerous small polygons which gives some insight into the distribution of such distances and the difficulty of such distance computations, and we analyze the distributions of the sizes of the largest and smaller resulting polygons.

scholarly journals Permutations Avoiding a Simsun Pattern

10.37236/9482 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Marilena Barnabei ◽  
Flavio Bonetti ◽  
Niccolò Castronuovo ◽  
Matteo Silimbani
Keyword(s):  
Main Tool ◽  
Binary Trees ◽  
Classical Sense ◽  
Set Of Permutations

A permutation $\pi$ avoids the simsun pattern $\tau$ if $\pi$ avoids the consecutive pattern $\tau$ and the same condition applies to  the restriction of $\pi$ to any interval $[k].$ Permutations avoiding the simsun pattern $321$ are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length $3$ in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length $3$ together with any set of classical patterns of length $3.$ The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees.

scholarly journals Grammaire de Ramanujan et Arbres de Cayley

10.37236/1275 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Dominique Dumont ◽  
Armand Ramamonjisoa
Keyword(s):  
Differential Operator ◽  
Rooted Trees ◽  
Combinatorial Interpretations

We study three sequences of polynomials defined as successive derivatives with respect to a differential operator associated with a grammar (one of these sequences was originally introduced by Ramanujan). Combinatorial interpretations for these polynomials are found in terms of rooted trees and graphs of mappings from $[n]$ to $[n]$.

scholarly journals Weighted partitions

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Rafael González S. D'León ◽  
Michelle L. Wachs
Keyword(s):  
Lie Algebra ◽  
Characteristic Polynomial ◽  
Rooted Trees ◽  
Free Lie Algebra ◽  
International Audience

International audience In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted <mathfrak>S</mathfrak>_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n.

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