scholarly journals Non-ambiguous trees: new results and generalization

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jean-Christophe Aval ◽  
Adrien Boussicault ◽  
Bérénice Delcroix-Oger ◽  
Florent Hivert ◽  
Patxi Laborde-Zubieta
Keyword(s):  
Binary Trees ◽  
Higher Dimension ◽  
International Audience ◽  
Definition Of ◽  
New Formula ◽  
Ential Equation

International audience We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differ- ential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain q-versions of our formula. And we generalize NATs to higher dimension.

scholarly journals On trees, tanglegrams, and tangled chains

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
Frderick Matsen IV
Keyword(s):  
Asymptotic Formula ◽  
Computer Science ◽  
Explicit Formula ◽  
Binary Trees ◽  
Biological Research ◽  
Open Problems ◽  
International Audience ◽  
Enumeration Formula ◽  
New Formula ◽  
Simple Asymptotic Formula

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

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.

New Similarity Measures between Vague Sets and Performance Analysis

2013 ◽  
Vol 811 ◽  
pp. 547-551 ◽  
Author(s):  
Hong Xu Wang ◽  
Hai Feng Wang ◽  
Kun Zhang ◽  
Hui Wang
Keyword(s):  
Data Mining ◽  
Performance Analysis ◽  
Similarity Measure ◽  
Similarity Measures ◽  
Practical Case ◽  
Vague Sets ◽  
Diagnosis Method ◽  
And Performance ◽  
Definition Of ◽  
New Formula

In order to amend the defects of existing similarity measure formula between vague sets, a new definition of similarity measure between vague sets is proposed and a new formula with higher resolution and highlighted uncertainty is presented on the basis of data mining vague value method. A general fault diagnosis method of Vague sets (GFDMVS) is proposed. The same practical case is studied with three methods and the results demonstrate the validity and reasonability of the method proposed in this paper.

scholarly journals A bijection between planar constellations and some colored Lagrangian trees

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Cedric Chauve
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Formula One ◽  
Lagrange Formula ◽  
International Audience ◽  
Planar Maps ◽  
New Formula

International audience Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, ...) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the partition of faces and vertices.

scholarly journals Recursions and divisibility properties for combinatorial Macdonald polynomials

2011 ◽  
Vol Vol. 13 no. 1 (Combinatorics) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Elizabeth Niese
Keyword(s):  
Hilbert Series ◽  
Permutation Statistics ◽  
Macdonald Polynomial ◽  
Integer Partition ◽  
Theoretical Methods ◽  
Major Index ◽  
International Audience ◽  
Definition Of ◽  
Fermionic Formulas ◽  
Divisibility Properties

Combinatorics International audience For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

scholarly journals Partitioned Cacti: a Bijective Approach to the Cycle Factorization Problem

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Gilles Schaeffer ◽  
Ekaterina Vassilieva
Keyword(s):  
Factorization Problem ◽  
Long Cycle ◽  
International Audience ◽  
Number Of Cycles ◽  
New Formula

International audience In this paper we construct a bijection for partitioned 3-cacti that gives raise to a new formula for enumeration of factorizations of the long cycle into three permutations with given number of cycles. Dans cet article, nous construisons une bijection pour 3-cacti partitionnés faisant apparaître une nouvelle formule pour l’énumération des factorisations d’un long cycle en trois permutations ayant un nombre donné de cycles.

scholarly journals Crystal energy via charge

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Anne Schilling
Keyword(s):  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Recursive Definition ◽  
R Matrix ◽  
Single Column ◽  
Nous Montrons ◽  
International Audience ◽  
Definition Of

International audience The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix. La formule de Ram et Yip pour les polynômes de Macdonald (à t = 0) fournit une statistique que nous appelons la charge. Dans les types ${A}$ et ${C}$, elle peut être définie sur les produits tensoriels des cristaux pour les colonnes de Kashiwara–Nakashima. Dans ce papier, nous montrons que la charge est égale à (l'opposé de) la fonction d'énergie sur cristaux affines. L'algorithme pour calculer la charge est bien plus simple que la définition récursive de l'énergie en fonction de la ${R}$-matrice combinatoire.

scholarly journals A Formula for the Möbius Function of the Permutation Poset Based on a Topological Decomposition

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jason P Smith
Keyword(s):  
Significant Proportion ◽  
Möbius Function ◽  
Permutation Patterns ◽  
Mobius Function ◽  
International Audience ◽  
Definition Of ◽  
Object Of Study ◽  
Pattern Containment ◽  
Exponential Complexity ◽  
Exponential Part

International audience The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.

scholarly journals On the analysis of ''simple'' 2D stochastic cellular automata

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Damien Regnault ◽  
Nicolas Schabanel ◽  
Eric Thierry
Keyword(s):  
Cellular Automata ◽  
Time Step ◽  
Energy Functions ◽  
Stochastic Cellular Automata ◽  
Von Neumann ◽  
Rich Variety ◽  
Moore Neighborhood ◽  
International Audience ◽  
Definition Of ◽  
New Phenomena

International audience Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but It is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007) This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions) Switching to the Moot e neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations) Nevertheless our methods allow to analyze different stages of the dynamics It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata

scholarly journals An explicit formula for ndinv, a new statistic for two-shuffle parking functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Angela Hicks ◽  
Yeonkyung Kim
Keyword(s):  
Explicit Formula ◽  
Parking Functions ◽  
Recursive Definition ◽  
Classical Definition ◽  
Inversion Number ◽  
International Audience ◽  
New Interpretation ◽  
The One ◽  
Definition Of ◽  
New Statistics

International audience In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, "ndinv'', on a family of parking functions. The definition was guided by a recursion satisfied by the polynomial $\langle\Delta_{h_m}C_p1C_p2...C_{pk}1,e_n\rangle$, for $\Delta_{h_m}$ a Macdonald eigenoperator, $C_{p_i}$ a modified Hall-Littlewood operator and $(p_1,p_2,\dots ,p_k)$ a composition of n. Using their new statistics, they are able to give a new interpretation for the polynomial $\langle\nabla e_n, h_j h_n-j\rangle$ as a q,t numerator of parking functions by area and ndinv. We recall that in the shuffle conjecture, parking functions are q,t enumerated by area and diagonal inversion number (dinv). Since their definition is recursive, they pose the problem of obtaining a non recursive definition. We solved this problem by giving an explicit formula for ndinv similar to the classical definition of dinv. In this paper, we describe the work we did to construct this formula and to prove that the resulting ndinv is the same as the one recursively defined by Duane, Garsia, and Zabrocki. Dans un travail récent Duane, Garsia et Zabrocki ont introduit une nouvelle statistique, "ndinv'' pour une famille de Fonctions Parking. Ce "ndinv" découle d'une récurrence satisfaite par le polynôme $\langle\Delta_{h_m}C_p1C_p2...C_{pk}1,e_n\rangle$, oú $\Delta_{h_m}$ est un opérateur linéaire avec fonctions propres les polynômes de Macdonald, les $C_{p_i}$ sont des opérateurs de Hall-Littlewood modifiés et $(p_1,p_2,\dots ,p_n)$ est un vecteur à composantes entières positives. Par moyen de cette statistique, ils ont réussi à donner une nouvelle interprétation combinatoire au polynôme $\langle\nabla e_n, h_j h_n-j\rangle$ on remplaçant "dinv'" par "ndinv". Rappelons nous que la conjecture "Shuffle"' exprime ce même polynôme comme somme pondérée de Fonctions Parking avec poids t à la "aire'" est q au "dinv". Puisque il donnent une définition récursive du "ndinv" il posent le problème de l'obtenir d'une façon directe. On rèsout se problème en donnant une formule explicite qui permet de calculer directement le "ndinv" à la manière de la formule classique du "dinv". Dans cet article on décrit le travail qu'on a fait pour construire cette formule et on démontre que nôtre formule donne le même "ndinv" récursivement construit par Duane, Garsia et Zabrocki.

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