scholarly journals The register function for lattice paths

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guy Louchard ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Lattice Paths ◽  
Binary Trees ◽  
International Audience

International audience The register function for binary trees is the minimal number of extra registers required to evaluate the tree. This concept is also known as Horton-Strahler numbers. We extend this definition to lattice paths, built from steps $\pm 1$, without positivity restriction. Exact expressions are derived for appropriate generating functions. A procedure is presented how to get asymptotics of all moments, in an almost automatic way; this is based on an earlier paper of the authors.

scholarly journals An extension to overpartitions of Rogers-Ramanujan identities for even moduli

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Jeremy Lovejoy ◽  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Lattice Paths ◽  
Infinite Products ◽  
International Audience

International audience We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.

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 $n$-color overpartitions, lattice paths, and multiple basic hypergeometric series

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Combinatorial Identities ◽  
Lattice Paths ◽  
Multiple Series ◽  
Infinite Products ◽  
Nous Montrons ◽  
International Audience ◽  
Multiple Basic Hypergeometric Series

International audience We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions. Nous définissons deux classes de séries hypergéométriques basiques multiples $V_{k,t}(a,q)$ et $W_{k,t}(a,q)$ qui généralisent des séries multiples étudiées par Agarwal, Andrews et Bressoud. Nous montrons comment interpréter ces séries comme les fonctions génératrices de chemins avec certaines restrictions et de surpartitions $n$-colorées vérifiant des conditions de différences pondérées. Nous remarquons aussi que certaines spécialisations de nos séries peuvent s'écrire comme des produits infinis, ce qui conduit à des identités combinatoires reliant les surpartitions $n$-colorées aux partitions ou surpartitions ordinaires.

scholarly journals Descendants and ascendants in binary trees

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Alois Panholzer ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Binary Tree ◽  
Binary Trees ◽  
Zeilberger’S Algorithm ◽  
International Audience ◽  
Explicit Formulae ◽  
Zeilberger's Algorithm

International audience There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (size of the tree) and j (a number between 1 and n), we consider the statistics number of ascendants of node j and number of descendants of node j. By appropriate trivariate generating functions, we are able to find explicit formulae for the expectation and the variance in all instances. The heavy computations that are necessary are facilitated by MAPLE and Zeilberger's algorithm. A similar problem comes fromlabelling the leaves from left to right by 1, 2, ..., n and considering the statistic number of ascendants (=height) of leaf j. For this, Kirschenhofer [1] has computed the average. With our approach, we are also able to get the variance. In the last section, a table with asymptotic equivalents is provided for the reader's convenience.

scholarly journals Generating functions for the area below some lattice paths

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Lattice Paths ◽  
The Matrix ◽  
International Audience ◽  
Riordan Array

International audience We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.

scholarly journals A triple product formula for plane partitions derived from biorthogonal polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Product Formula ◽  
Triple Product ◽  
Lattice Paths ◽  
Plane Partitions ◽  
International Audience ◽  
Biorthogonal Polynomials ◽  
Bounded Size

International audience A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.

scholarly journals Enumeration of Binary Trees and Universal Types

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Generating Functions ◽  
Path Length ◽  
Empirical Distribution ◽  
Applied Mathematics ◽  
Binary Trees ◽  
Wkb Method ◽  
Interesting Problem ◽  
Matched Asymptotics ◽  
Ordered Trees ◽  
International Audience

International audience Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a \emphgiven path length (sum of depths) are there? This question arose in the study of \emphuniversal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, T_p, of given path length p (and also the number of distinct Lempel-Ziv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to T_p ~ 2^2p/(log_2 p)(1+O(log ^-2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics.

scholarly journals The b-chromatic number of power graphs

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Brice Effantin ◽  
Hamamache Kheddouci
Keyword(s):  
Chromatic Number ◽  
Binary Trees ◽  
Proper Coloring ◽  
Power Cycles ◽  
International Audience

International audience The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

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 Random sampling of lattice paths with constraints, via transportation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Lucas Gerin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Sampling ◽  
Optimal Transport ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Contraction Property ◽  
International Audience

International audience We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.

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