scholarly journals Permutations avoiding an increasing number of length-increasing forbidden subsequences

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Elena Barcucci ◽  
Alberto Del Lungo ◽  
Elisa Pergola ◽  
Renzo Pinzani
Keyword(s):  
Generating Function ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Pairwise Comparisons ◽  
Number Sequence ◽  
Schröder Numbers ◽  
International Audience

International audience A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequences of the form σ (j+1)(j+2), σ being any permutation on \1,...,j\. For j=1 the only subsequence in F^1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F^2 are 1234 2134 and the (1234,2134)avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F^j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence \a_n\ such that C_n ≤ a_n ≤ n!, C_n being the nth Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F^j according to the length, to the number of left minima and of non-inversions.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t$-Catalan numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li
Keyword(s):  
Vector Space ◽  
Hilbert Series ◽  
Upper Bounds ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Nous Obtenons ◽  
Minimal Set ◽  
Nous Trouvons ◽  
International Audience

International audience Let $I_n$ be the (big) diagonal ideal of $(\mathbb{C}^2)^n$. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ spanned by a minimal set of generators for $I_n$. We give simple upper bounds on $\textrm{dim} (M_n)_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\textrm{dim} (M_n)_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $(M_n)_{d_1, d_2}$. Soit $I_n$ l'idéal de la (grande) diagonale de $(\mathbb{C}^2)^n$. Haiman a démontré que le $q,t$-nombre de Catalan est la série de Hilbert de l'espace vectoriel gradué $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ engendré par un ensemble minimal de générateurs de $I_n$. Nous obtenons des bornes supérieures simples pour $\textrm{dim} (M_n)_{d_1, d_2}$ en termes de nombres de partitions, ainsi que tous les bi-degrés $(d_1, d_2)$ pour lesquels ces bornes supérieures sont atteintes. Pour ces bi-degrés, nous trouvons aussi des bases explicites de $(M_n)_{d_1, d_2}$.

scholarly journals Bicoloured Dyck Paths and the Contact Polynomial for $n$ Non-Intersecting Paths in a Half-Plane Lattice

10.37236/1728 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
R. Brak ◽  
J. W. Essam
Keyword(s):  
Recurrence Relation ◽  
Half Plane ◽  
Generating Function ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Lattice Paths ◽  
Product Form ◽  
Dyck Path ◽  
Combinatorial Interpretation ◽  
Dyck Paths

In this paper configurations of $n$ non-intersecting lattice paths which begin and end on the line $y=0$ and are excluded from the region below this line are considered. Such configurations are called Hankel $n-$paths and their contact polynomial is defined by $\hat{Z}^{\cal{H}}_{2r}(n;\kappa)\equiv \sum_{c= 1}^{r+1} |{\cal H}_{2r}^{(n)}(c)|\kappa^c$ where ${\cal H}_{2r}^{(n)}(c)$ is the set of Hankel $n$-paths which make $c$ intersections with the line $y=0$ the lowest of which has length $2r$. These configurations may also be described as parallel Dyck paths. It is found that replacing $\kappa$ by the length generating function for Dyck paths, $\kappa(\omega) \equiv \sum_{r=0}^\infty C_r \omega^r$, where $C_r$ is the $r^{th}$ Catalan number, results in a remarkable simplification of the coefficients of the contact polynomial. In particular it is shown that the polynomial for configurations of a single Dyck path has the expansion $\hat{Z}^{\cal{H}}_{2r}(1;\kappa(\omega)) = \sum_{b=0}^\infty C_{r+b}\omega^b$. This result is derived using a bijection between bi-coloured Dyck paths and plain Dyck paths. A bi-coloured Dyck path is a Dyck path in which each edge is coloured either red or blue with the constraint that the colour can only change at a contact with the line $y=0$. For $n>1$, the coefficient of $\omega^b$ in $\hat{Z}^{\cal{W}}_{2r}(n;\kappa(\omega))$ is expressed as a determinant of Catalan numbers which has a combinatorial interpretation in terms of a modified class of $n$ non-intersecting Dyck paths. The determinant satisfies a recurrence relation which leads to the proof of a product form for the coefficients in the $\omega$ expansion of the contact polynomial.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals An integral representation, complete monotonicity, and inequalities of the Catalan numbers

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Fang-Fang Liu
Keyword(s):  
Integral Representation ◽  
Generating Function ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Cauchy Integral ◽  
Complex Functions

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

Closed forms for convolutions of Catalan numbers

2012 ◽  
Vol 96 (535) ◽  
pp. 82-90 ◽  
Author(s):  
N. Gauthier
Keyword(s):  
Natural Number ◽  
Catalan Numbers ◽  
Number Sequence ◽  
Counting Problems

In Catalan numbers with applications [1], Thomas Koshy gives a clear and accessible account of some of the important characteristics of Catalan numbers. The author also discusses the significance of these numbers for mathematicians, scientists, engineers and amateur enthusiasts alike. The general contents of his book are reviewed in [2]. Catalan numbers, which are named after the mathematician Eugene Catalan (1814-1894), have very interesting and sometimes surprising properties.Catalan numbers occur in combinatorial mathematics, in particular, where they show up as a natural number sequence that arises in counting problems that usually involve recursively-defined objects.

scholarly journals On powers of the Catalan number sequence

2019 ◽  
Vol 342 (7) ◽  
pp. 2139-2147
Author(s):  
Gwo Dong Lin
Keyword(s):  
Catalan Number ◽  
Number Sequence

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

Sign in / Sign up

Export Citation Format

Share Document