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 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 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.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

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 The height of random binary unlabelled trees

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Nicolas Broutin ◽  
Philippe Flajolet
Keyword(s):  
Large Deviations ◽  
Complex Plane ◽  
Generating Functions ◽  
Binary Trees ◽  
International Audience ◽  
Large Deviations Estimates

International audience This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.

scholarly journals New hook length formulas for binary trees

COMBINATORICA ◽  
2010 ◽  
Vol 30 (2) ◽  
pp. 253-256 ◽  
Author(s):  
Guo-Niu Han
Keyword(s):  
Binary Trees ◽  
Hook Length

scholarly journals Bootstrapping and double-exponential limit laws

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Maximum Degree ◽  
Exponential Rate ◽  
Limit Laws ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
Longest Run ◽  
Plane Trees ◽  
International Audience ◽  
Motzkin Paths

Combinatorics International audience We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.

scholarly journals q-Hermite Polynomials and Classical Orthogonal Polynomials

1996 ◽  
Vol 48 (1) ◽  
pp. 43-63 ◽  
Author(s):  
Christian Berg ◽  
Mourad E. H. Ismail
Keyword(s):  
Weight Function ◽  
Generating Functions ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Rational Functions ◽  
Classical Orthogonal Polynomials ◽  
Beta Integral ◽  
Orthogonality Relations ◽  
Beta Integrals ◽  
Wilson Polynomials

AbstractWe use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

scholarly journals Equipopularity Classes of 132-Avoiding Permutations

10.37236/3675 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Lynn Chua ◽  
Krishanu Roy Sankar
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
Connected Components ◽  
Plane Tree ◽  
Plane Trees ◽  
Set Of Permutations ◽  
Spine Structure

The popularity of a pattern $p$ in a set of permutations is the sum of the number of copies of $p$ in each permutation of the set. We study pattern popularity in the set of 132-avoiding permutations. Two patterns are equipopular if, for all $n$, they have the same popularity in the set of length-$n$ 132-avoiding permutations. There is a well-known bijection between 132-avoiding permutations and binary plane trees. The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines. Rudolph shows that patterns of the same length are equipopular if their associated binary plane trees have the same spine structure. We prove the converse of this result using the method of generating functions, which gives a complete classification of 132-avoiding permutations into equipopularity classes.

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