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 Hook Length Formulas for Trees by Han's Expansion

10.37236/151 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
William Y.C. Chen ◽  
Oliver X.Q. Gao ◽  
Peter L. Guo
Keyword(s):  
Weight Function ◽  
General Formula ◽  
Generating Functions ◽  
Binary Trees ◽  
Hook Length ◽  
Labeled Trees ◽  
Plane Trees

Recently Han obtained a general formula for the weight function corresponding to the expansion of a series in terms of hook lengths of binary trees. In this paper, we present weight function formulas for $k$-ary trees, plane trees, plane forests, labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu, Han, Gessel and Seo, and Postnikov.

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 Hopf algebras on planar trees and permutations

2021 ◽  
Author(s):  
Diego Arcis ◽  
Sebastián Márquez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Labeled Trees ◽  
Different Types ◽  
Structure Lead ◽  
Planar Rooted Trees

We endow the space of rooted planar trees with the structure of a Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labeled trees, [Formula: see text]-trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto–Reutenauer and Loday–Ronco via planar rooted trees.

scholarly journals Toward a Combinatorial Proof of the Jacobian Conjecture!

10.37236/2023 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Dan Singer
Keyword(s):  
Vector Space ◽  
Full Rank ◽  
Coefficient Matrix ◽  
Rooted Trees ◽  
Jacobian Conjecture ◽  
Combinatorial Proof ◽  
Combinatorial Properties ◽  
Linearly Independent ◽  
Acyclic Digraph ◽  
Fixed Tree

The Jacobian conjecture [Keller, Monatsh. Math. Phys., 1939] gives rise to a problem in combinatorial linear algebra: Is the vector space generated by rooted trees spanned by forest shuffle vectors? In order to make headway on this problem we must study the algebraic and combinatorial properties of rooted trees. We prove three theorems about the vector space generated by binary rooted trees: Shuffle vectors of fixed length forests are linearly independent, shuffle vectors of nondegenerate forests relative to a fixed tree are linearly independent, and shuffle vectors of sufficient length forests are linearly independent. These results are proved using the acyclic digraph method for establishing that a coefficient matrix has full rank [Singer, The Electronic Journal of Combinatorics, 2009]. We also provide an infinite class of counterexamples to demonstrate the need for sufficient length in the third theorem.

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 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 The Cambrian Hopf Algebra

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
G. Chatel ◽  
V. Pilaud
Keyword(s):  
Hopf Algebra ◽  
Binary Search ◽  
Binary Trees ◽  
Search Trees ◽  
Local Conditions ◽  
Bijective Correspondence ◽  
Combinatorial Properties ◽  
Labeled Trees ◽  
Signed Permutations ◽  
International Audience

International audience Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday and M. Ronco’s algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. Finally, we define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements. Les arbres Cambriens sont des arbres orientés et étiquetés qui satisfont des conditions locales autour de leurs nœuds généralisant les conditions des arbres binaires de recherche classiques. A partir de la correspondence bijective entre permutations signées et arbres Cambriens à niveau, nous définissons l’algèbre Cambrienne qui généralise l’algèbre sur les arbres binaires de J.-L. Loday et M. Ronco. Nous donnons une description combinatoire du produit et du coproduit aussi bien dans l’algèbre Cambrienne que dans sa duale en termes d’opérations sur les arbres Cambriens. Enfin, nous définissons des bases multiplicatives de l’algèbre Cambrienne et étudions les propriétés structurelles et énumératives de leurs éléments indécomposables.

scholarly journals On Symmetries in Phylogenetic Trees

10.37236/5994 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Éric Fusy
Keyword(s):  
Simple Formula ◽  
Phylogenetic Trees ◽  
Binary Trees ◽  
Combinatorial Proof ◽  
Random Generation ◽  
Integer Partition ◽  
Binary Partition ◽  
Generation Procedure

Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\geq 1$ leaves, fixed (for the relabelling action) by a given permutation $\sigma\in\frak{S}_n$. Denoting by $\lambda\vdash n$ the integer partition giving the sizes of the cycles of $\sigma$ in non-increasing order, they show by a guessing/checking approach that if $\lambda$ is a binary partition (it is known that $a_n(\sigma)=0$ otherwise), then$$a_n(\sigma)=\prod_{i=2}^{\ell(\lambda)}(2(\lambda_i+\cdots+\lambda_{\ell(\lambda)})-1),$$and they derive from it a formula and random generation procedure for tanglegrams (and more generally for tangled chains). Our main result is a combinatorial proof of the formula for $a_n(\sigma)$, which yields a simplification of the random sampler for tangled chains.

scholarly journals Bruhat Order on Fixed-Point-Free Involutions in the Symmetric Group

10.37236/3861 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Matthew Watson
Keyword(s):  
Fixed Point ◽  
Symmetric Group ◽  
Bruhat Order ◽  
Binary Trees ◽  
Combinatorial Proof ◽  
Combinatorial Identity ◽  
Structural Description ◽  
Rook Placements ◽  
Dyck Paths ◽  
New Interpretation

We provide a structural description of Bruhat order on the set $F_{2n}$ of fixed-point-free involutions in the symetric group $S_{2n}$ which yields a combinatorial proof of a combinatorial identity that is an expansion of its rank-generating function. The decomposition is accomplished via a natural poset congruence, which yields a new interpretation and proof of a combinatorial identity that counts the number of rook placements on the Ferrers boards lying under all Dyck paths of a given length $2n$. Additionally, this result extends naturally to prove new combinatorial identities that sum over other Catalan objects: 312-avoiding permutations, plane forests, and binary trees.

scholarly journals Superization and $(q,t)$-Specialization in Combinatorial Hopf Algebras

10.37236/87 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Jean-Yves Thibon
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions ◽  
Binary Trees ◽  
Hook Length ◽  
Combinatorial Hopf Algebras ◽  
Plane Trees ◽  
Classical Construction

We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hook-content formula for the $(q,t)$-specializations of various bases. Exploiting the dendriform structures yields in particular $(q,t)$-analogs of the Björner-Wachs $q$-hook-length formulas for binary trees, and similar formulas for plane trees.

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