scholarly journals Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

2012 ◽  
Vol Vol. 14 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris ◽  
Karen A. Yeats
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Generating Functions ◽  
Second Order ◽  
Radius Of Convergence ◽  
Structure Theory ◽  
Strengthened Version ◽  
International Audience

Automata, Logic and Semantics International audience Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.

scholarly journals The enumeration of fully commutative affine permutations

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Christopher R. H. Hanusa ◽  
Brant C. Jones
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Nous Obtenons ◽  
Full Version ◽  
International Audience ◽  
Affine Permutations ◽  
Fixed Rank

International audience We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. This is a summary of the results; the full version appears elsewhere. Nous présentons une fonction génératrice qui énumère les permutations affines totalement commutatives par leur rang et par leur longueur de Coxeter, généralisant les formules dues à Stembridge et à Barcucci–Del Lungo–Pergola–Pinzani. Pour un rang précis, les fonctions génératrices ont des coefficients qui sont périodiques de période divisant leur rang. Nous obtenons des résultats qui expliquent la structure des permutations affines totalement commutatives. L'article dessous est un aperçu des résultats; la version complète appara\^ıt ailleurs.

scholarly journals Counting words with Laguerre polynomials

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Equal Length ◽  
Combinatorial Interpretation ◽  
International Audience ◽  
Nous Utilisons

International audience We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for $k$-ary words avoiding any vincular pattern that has only ones. We also give generating functions for $k$-ary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions. Nous développons une méthode pour compter des mots satisfaisants certaines restrictions en établissant une interprétation combinatoire utile d’un produit de sommes formelles de polynômes de Laguerre. Nous utilisons cette méthode pour trouver la série génératrice pour les mots $k$-aires évitant les motifs vinculars consistant uniquement de uns. Nous présentons en suite les séries génératrices pour les mots $k$-aires évitant de façon cyclique les motifs vinculars consistant uniquement de uns et dont chaque série de uns entre deux tirets est de la même longueur. Nous présentons aussi les résultats analogues pour les compositions.

scholarly journals Rhombic alternative tableaux, assemblees of permutations, and the ASEP

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olya Mandelshtam ◽  
Xavier Viennot
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Generating Functions ◽  
Finite Lattice ◽  
One Dimensional ◽  
Open Boundaries ◽  
Bijective Proof ◽  
Insertion Algorithm ◽  
International Audience ◽  
Steady State Probabilities

International audience In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.

scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

scholarly journals Avoiding maximal parabolic subgroups of S_k

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Toufik Mansour ◽  
Alek Vainshtein
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Parabolic Subgroups ◽  
International Audience ◽  
Maximal Parabolic Subgroups

International audience We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.

scholarly journals Computing generating functions of ordered partitions with the transfer-matrix method

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Masao Ishikawa ◽  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Transfer Matrix ◽  
Generating Function ◽  
Generating Functions ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Stirling Number ◽  
International Audience

International audience An ordered partition of $[n]:=\{1,2,\ldots, n\}$ is a sequence of disjoint and nonempty subsets, called blocks, whose union is $[n]$. The aim of this paper is to compute some generating functions of ordered partitions by the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson, which assert that the generating function of some statistics of ordered partitions give rise to a natural $q$-analogue of $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind.

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 When Structures Are Almost Surely Connected

10.37236/1514 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Jason P. Bell
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Radius Of Convergence

Let $A_n$ denote the number of objects of some type of"size" $n$, and let $C_n$ denote the number of these objects which are connected. It is often the case that there is a relation between a generating function of the $C_n$'s and a generating function of the $A_n$'s. Wright showed that if $\lim_{n\rightarrow\infty} C_n/A_n =1$, then the radius of convergence of these generating functions must be zero. In this paper we prove that if the radius of convergence of the generating functions is zero, then $\limsup_{n\rightarrow \infty} C_n/A_n =1$, proving a conjecture of Compton; moreover, we show that $\liminf_{n\rightarrow\infty} C_n/A_n$ can assume any value between $0$ and $1$.

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 Degree distribution in random planar graphs

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Omer Gimenez ◽  
Marc Noy
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Degree Distribution ◽  
Planar Graphs ◽  
Probability Generating Function ◽  
Asymptotic Estimates ◽  
Asymptotic Enumeration ◽  
Root Vertex ◽  
International Audience ◽  
Transfer Theorems

International audience We prove that for each $k \geq 0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimènez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.

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