scholarly journals Four Classes of Pattern-Avoiding Permutations Under One Roof: Generating Trees with Two Labels

10.37236/1691 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Functional Equations ◽  
Tutte Polynomial ◽  
Motzkin Numbers ◽  
Generating Trees ◽  
Generating Tree ◽  
Planar Maps ◽  
Bivariate Generating Function ◽  
Pattern Avoiding Permutations

Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of $123$-avoiding permutations. The rewriting rule automatically gives a functional equation satisfied by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series. Several other families of permutations can be described by a generating tree in which each node carries two integer labels. To these trees correspond other functional equations, defining 3-variate generating functions. We propose an approach to solving such equations. We thus recover and refine, in a unified way, some results on Baxter permutations, $1234$-avoiding permutations, $2143$-avoiding (or: vexillary) involutions and $54321$-avoiding involutions. All the generating functions we obtain are D-finite, and, more precisely, are diagonals of algebraic series. Vexillary involutions are exceptionally simple: they are counted by Motzkin numbers, and thus have an algebraic generating function. In passing, we exhibit an interesting link between Baxter permutations and the Tutte polynomial of planar maps.

scholarly journals Generating trees for partitions and permutations with no k-nestings

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Sergi Elizalde ◽  
Marni Mishna ◽  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Young Tableaux ◽  
Set Partitions ◽  
Lattice Walks ◽  
Generating Trees ◽  
Generating Tree ◽  
International Audience ◽  
Tree Approach

International audience We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Nous décrivons une approche, basée sur l'utilisation d'arbres de génération, pour énumération et la génération exhaustive de partitions et permutations sans k-emboîtement. Contrairement aux travaux antérieurs qui reposent sur un lien entre ces objets, tableaux de Young et familles de chemins dans des treillis, notre approche traite directement partitions et diagrammes de permutations. Nous fournissons des équations fonctionnelles explicites pour les séries génératrices, avec k en tant que paramètre.

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 Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

scholarly journals Continued Fractions and Catalan Problems

10.37236/1523 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Mahendra Jani ◽  
Robert G. Rieper
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Ordered Trees ◽  
Pattern Avoiding Permutations ◽  
Ordered Tree ◽  
Different Levels

We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of $(132)$-pattern avoiding permutations that have a given number of increasing patterns of length $k$. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case $k=3$.

scholarly journals The Double Riordan Group

10.37236/2034 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Dennis E. Davenport ◽  
Louis W. Shapiro ◽  
Leon C. Woodson
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Triangular Matrices ◽  
Ordered Trees ◽  
Dyck Paths ◽  
Motzkin Numbers ◽  
The Matrix ◽  
Riordan Group ◽  
Lower Triangular ◽  
Lower Triangular Matrices

The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, $g$ and $f$. The kth column of the matrix has the generating function $gf^k$. In the Double Riordan group there are two generating function $f_1$ and $f_2$ such that the columns, starting at the left, have generating functions using $f_1$ and $f_2$ alternately. Examples include Dyck paths with level steps of length 2  allowed at even height and also ordered trees with differing degree possibilities at even and odd height(perhaps representing summer and winter). The Double Riordan group is a generalization not of the Riordan group itself but of the checkerboard subgroup. In this context both familiar and far less familiar sequences occur such as the Motzkin numbers and the number of spoiled child trees. The latter is a slightly enhanced cousin of ordered trees which are counted by the Catalan numbers.

scholarly journals A Note on Irreducible Maps with Several Boundaries

10.37236/3443 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
J. Bouttier ◽  
E. Guitter
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Planar Maps ◽  
Number Of Faces ◽  
Multiple Edges ◽  
Explicit Expressions

We derive a formula for the generating function of $d$-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which is recovered by taking $d=0$. As an application, we obtain an expression for the number of $d$-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges ($d=2$), $4$-irreducible maps and maps of girth at least $6$ ($d=4$). Our derivation is based on a tree interpretation of the various encountered generating functions.

scholarly journals On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers

2019 ◽  
Vol 13 (2) ◽  
pp. 478-494 ◽  
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Special Numbers And Polynomials ◽  
Combinatorial Sums ◽  
Finite Sums ◽  
Computation Algorithms

In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for these numbers and polynomials. By using these algorithms, we provide several values of these numbers and polynomials. Furthermore, some new identities, formulas and combinatorial sums are obtained by using relations derived from the functional equations of these generating functions. These identities and formulas include the Apostol-type numbers and polynomials, and also the Stirling numbers. Finally, we give further remarks and observations on the generating function including ?-Apostol-Daehee numbers, special numbers, and finite sums.

scholarly journals Three-Letter-Pattern Avoiding Permutations and Functional Equations

10.37236/1077 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ghassan Firro ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Recurrence Relations ◽  
System Of Functional Equations ◽  
Pattern Avoiding Permutations ◽  
Letter Pattern

We present an algorithm for finding a system of recurrence relations for the number of permutations of length $n$ that satisfy a certain set of conditions. A rewriting of these relations automatically gives a system of functional equations satisfied by the multivariate generating function that counts permutations by their length and the indices of the corresponding recurrence relations. We propose an approach to describing such equations. In several interesting cases the algorithm recovers and refines, in a unified way, results on $\tau$-avoiding permutations and permutations containing $\tau$ exactly once, where $\tau$ is any classical, generalized, and distanced pattern of length three.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

Sign in / Sign up

Export Citation Format

Share Document