scholarly journals Some results for monotonically labelled simply generated trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger ◽  
Alois Panholzer
Keyword(s):  
Tree Size ◽  
Random Node ◽  
International Audience ◽  
Simply Generated Trees ◽  
Special Case

International audience We consider simply generated trees, where the nodes are equipped with weakly monotone labellings with elements of $\{1, 2, \ldots, r\}$, for $r$ fixed. These tree families were introduced in Prodinger and Urbanek (1983) and studied further in Kirschenhofer (1984), Blieberger (1987), and Morris and Prodinger (2005). Here we give distributional results for several tree statistics (the depth of a random node, the ancestor-tree size and the Steiner-distance of $p$ randomly chosen nodes, the height of the $j$-st leaf, and the number of nodes with label $l$), which extend the existing results and also contain the corresponding results for unlabelled simply generated trees as the special case $r=1$.

scholarly journals Enumerating alternating tree families

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Nous Obtenons ◽  
Asymptotic Results ◽  
Root Node ◽  
Ordered Trees ◽  
Exponential Generating Function ◽  
Random Node ◽  
Enumeration Problems ◽  
International Audience ◽  
Nous Examinons ◽  
Labelled Trees

International audience We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results. Nous étudions deux problèmes de dénombrement d'$\textit{arbres alternés haut-bas}$ : par définition, ce sont des arbres munis d'une racine et tels que, pour tout chemin partant de la racine, les valeurs $v_1,v_2,v_3,\ldots$ associées aux nœuds du chemin satisfont la chaîne d'inégalités $v_1 < v_2 > v_3 < v_4 > \cdots$. D'une part, nous considérons diverses familles d'arbres intéressantes du point de vue de l'analyse combinatoire (comme les arbres de Motzkin, les arbres non ordonnés, ordonnés et $d$-aires) et nous étudions pour chaque famille le nombre total $T_n$ d'arbres alternés haut-bas de taille $n$. Nous obtenons pour toutes les familles d'arbres considérées une caractérisation implicite de la fonction génératrice exponentielle $T(z)$. Cette caractérisation nous renseigne sur le comportement asymptotique des coefficients $T_n$ de plusieurs familles d'arbres. D'autre part, nous examinons le cas particulier de la famille des arbres ordonnés : nous étudions l'influence de l'étiquetage alterné haut-bas sur l'allure générale de ces arbres en analysant trois paramètres dans un arbre aléatoire (valeur de la racine, degré de la racine et profondeur d'un nœud aléatoire). Nous obtenons alors des résultats en terme de distribution limite, mais aussi de dénombrement exact.

scholarly journals The profile of unlabeled trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger
Keyword(s):  
Local Time ◽  
Joint Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion ◽  
Level Number ◽  
International Audience ◽  
The Times ◽  
Simply Generated Trees

International audience We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

scholarly journals On the monotone hook hafnian conjecture

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Mirkó Visontai
Keyword(s):  
Graph Polynomials ◽  
Real Roots ◽  
Threshold Graphs ◽  
Multivariate Generalization ◽  
International Audience ◽  
Special Case

International audience We investigate a conjecture of Haglund that asserts that certain graph polynomials have only real roots. We prove a multivariate generalization of this conjecture for the special case of threshold graphs. Nous étudions une conjecture de Haglund qui affirme que certaines polynômes des graphes ont uniquement des racines réelles. Nous prouvons une généralisation multivariée de cette conjecture pour le cas particulier des graphes à seuil.

scholarly journals Enumerating (2+2)-free posets by the number of minimal elements and other statistics

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Generating Function ◽  
Minimal Elements ◽  
Nous Montrons ◽  
International Audience ◽  
Ascent Sequences ◽  
Special Case

International audience A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements. Un poset sera dit (2+2)-libre s'il ne contient aucun sous-poset isomorphe à 2+2, l'union disjointe de deux chaînes à deux éléments. Dans un article récent, Bousquet-Mélou et al. ont trouvé, à l'aide de "suites de montées'', la fonction génératrice des nombres de posets (2+2)-libres: c'est $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. Nous étendons ce résultat en trouvant la fonction génératrice des posets (\textrm2+2)-libres rendant compte de quatre statistiques, dont le nombre d'éléments minimaux du poset. Nous montrons aussi que lorsqu'on ne s'intéresse qu'au nombre d'éléments minimaux, notre fonction génératrice assez compliquée peut être simplifiée en$P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$, où $p_n,k$ est le nombre de posets (2+2)-libres de taille $n$ avec $k$ éléments minimaux.

scholarly journals On the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals On the class number formula of certain real quadratic fields

2013 ◽  
Vol Volume 36 ◽  
Author(s):  
K Chakraborty ◽  
S Kanemitsu ◽  
T Kuzumaki
Keyword(s):  
Class Number ◽  
Character Sums ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
International Audience ◽  
Special Case ◽  
Real Quadratic

International audience In this note we give an alternate expression of class number formula for real quadratic fields with discriminant $d \equiv 5\, {\rm mod}\, 8$. %Dirichlet's classical class number formula for real quadratic fields expresses `class number' in somewhat `transcend' manner, which was simplified by P. Chowla in the special case when the discriminant $d = p \equiv 5\,{\rm mod}\, 8$ is a prime. We use another form of class number formula and transform it using Dirichlet's $1/4$-th character sums. Our result elucidates other generalizations of the class number formula by Mitsuhiro, Nakahara and Uhera for general real quadratic fields.

scholarly journals Dual combinatorics of zonal polynomials

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Irreducible Character ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Combinatorial Formula ◽  
Zonal Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Recent Developments ◽  
International Audience ◽  
Special Case

International audience In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

Sign in / Sign up

Export Citation Format

Share Document