scholarly journals Sorting with Pattern-Avoiding Stacks: The 132-Machine

10.37236/9642 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Giulio Cerbai ◽  
Anders Claesson ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Lattice Paths ◽  
Open Problems ◽  
Set Partitions ◽  
Sort It ◽  
Combinatorial Objects ◽  
Growth Functions ◽  
Set Of Permutations

This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed by the above mentioned authors. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.

Explicit Limits of Total Variation Distance in Approximations of Random Logarithmic Assemblies by Related Poisson Processes

1997 ◽  
Vol 6 (1) ◽  
pp. 87-105 ◽  
Author(s):  
DUDLEY STARK
Keyword(s):  
Total Variation ◽  
Local Limit ◽  
Total Variation Distance ◽  
Weighted Sums ◽  
Set Partitions ◽  
Ewens Sampling Formula ◽  
Combinatorial Objects ◽  
Sampling Formula ◽  
Random Mappings ◽  
Variation Distance

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Zi conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which i[ ]Zi → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β ε (0, 1]. For logarithmic assemblies, the total variation distance db(n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β ε (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that db(n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavaré for the Ewens sampling formula. Local limit theorems for weighted sums of the Zi are used to prove these results.

scholarly journals Arc-Coloured Permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Joint Distribution ◽  
Rational Functions ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Crossing Numbers ◽  
Rational Series ◽  
Nous Montrons ◽  
International Audience ◽  
Labelled Graphs

International audience The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. <begin>otherlanguage*</begin>french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. <end>otherlanguage*</end>

scholarly journals The dominance order for permutations

2015 ◽  
Vol 25 (1) ◽  
pp. 45-62
Author(s):  
Niccoló Castronuovo
Keyword(s):  
Order Relation ◽  
Lattice Paths ◽  
Dominance Order ◽  
Young Tableaux ◽  
Order Relations ◽  
Order Isomorphisms ◽  
Set Of Permutations

Abstract We define an order relation over Sn considering the Robinson-Schensted bijection and the dominance order over Young tableaux. This order relation makes Sn(k k - 1...3 2 1) -the set of permutations of length n that avoid the pattern k k - 1...3 2 1, k ≤ n- a principal filter in Sn. We study in detail these order relations on Sn(321) and Sn(4321), finding order-isomorphisms between these sets and sets of lattice paths.

scholarly journals Crossings and Nestings for Arc-Coloured Permutations and Automation

10.37236/4080 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Joint Distribution ◽  
Rational Functions ◽  
Set Partitions ◽  
Combinatorial Objects

Symmetric joint distribution between crossings and nestings was established in several combinatorial objects. Recently, Marberg extended Chen and Guo's result on coloured matchings to coloured set partitions following a multi-dimensional generalization of the bijection and enumerative methods from Chen, Deng, Du, Stanley, and Yan. We complete the study for arc-coloured permutations by establishing symmetric joint distribution for crossings and nestings and by showing that the ordinary generating functions for $j$-noncrossing, $k$-nonnesting, $r$-coloured permutations according to size $n$ are rational functions. Finally, we automate the generation of these rational functions and analyse the first $70$ series.

scholarly journals On Parking Functions and the Zeta Map in Types B, C and D

10.37236/6714 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Hilbert Series ◽  
Root Systems ◽  
The Other ◽  
Lattice Paths ◽  
Square Lattice ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Coxeter Number ◽  
Dyck Paths ◽  
Diagonal Harmonics

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

scholarly journals Generating Random Elements of Finite Distributive Lattices

10.37236/1330 ◽  
1997 ◽  
Vol 4 (2) ◽  
Author(s):  
James Propp
Keyword(s):  
Lattice Paths ◽  
Distributive Lattices ◽  
Alternating Sign Matrices ◽  
Combinatorial Objects ◽  
Survey Article ◽  
The Past ◽  
Coupling From The Past ◽  
Finite Set ◽  
Monte Carlo Randomization ◽  
Finite Distributive Lattices

This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using "coupling from the past" to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, an alternating sign matrices.

scholarly journals Extending from bijections between marked occurrences of patterns to all occurrences of patterns

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jeffrey Remmel ◽  
Mark Tiefenbruck
Keyword(s):  
Nous Avons ◽  
Large Class ◽  
Permutation Patterns ◽  
Open Problems ◽  
Combinatorial Objects ◽  
International Audience ◽  
General Method

International audience We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results. Nous considérons deux derniers problèmes ouverts indiquant que certaines statistiques sur les divers ensembles d'objets combinatoires sont équiréparties. La première, posée par Anders Claesson et Svante Linusson, concerne les imbrications dans des filtrages sur $\{1,2,\ldots,2n\}$ pour les occurrences d'un certain modèle de permutations dans $S_n$. La seconde, posée par Miles Jones et Jeffrey Remmel, concerne les occurrences d'une large classe de schémas de permutation consécutive aux évènements du même modèle dans les cycles de permutations. Nous développons une méthode générale qui résout ces deux problèmes et beaucoup plus. Nous avons également utiliser le principe d'involution Garsia-Milne pour obtenir des preuves purement bijectives de ces résultats.

scholarly journals Rook Theory, Generalized Stirling Numbers and $(p,q)$-Analogues

10.37236/1837 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
J. B. Remmel ◽  
Michelle L. Wachs
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Set Partitions ◽  
Growth Functions ◽  
Rook Theory ◽  
Signed Permutations ◽  
Generalized Stirling Numbers ◽  
Alpha Beta ◽  
Natural Statistics ◽  
Appl Math

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

Total Variation Asymptotics for Refined Poisson Process Approximations of Random Logarithmic Assemblies

1999 ◽  
Vol 8 (6) ◽  
pp. 567-598 ◽  
Author(s):  
DUDLEY STARK
Keyword(s):  
Poisson Process ◽  
Total Variation ◽  
Counting Process ◽  
Regular Graphs ◽  
Total Size ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Sampling Formula ◽  
Random Mappings ◽  
Variation Distance

Assemblies are decomposable combinatorial objects characterized by a sequence mi that counts the number of possible components of size i. Permutations on n elements, mappings from a set containing n elements into itself, 2-regular graphs on n vertices, and set partitions on a set of size n are all assemblies with natural decompositions. Logarithmic assemblies are characterized by constants θ > 0 and κ0 > 0 such that miκi0/ (i−1)! → θ. Random mappings, permutations and 2-regular graphs are all logarithmic assemblies, but set partitions are not.Given a logarithmic assembly, all representatives having total size n are chosen uniformly and a component counting process C(n) = (C1(n), C2(n), …, Cn(n)) is defined, where Ci(n) is the number of components of size i. Our results also apply to C(n) distributed as the Ewens sampling formula with parameter θ. Denote the component counting process up to size at most b by Cb(n) = (C1(n), C2(n), …, Cb(n)). It is natural to approximate Cb by Zb = (Z1, Z2, …, Zb), the b-dimensional process of independent Poisson variables Zi for which the ith variable has expectation [ ]Zi = miκi0 exp((1−θ)i/n)/i!. We find asymptotics for the total variation distance between Cb(n) and Zb.

scholarly journals Two by two squares in set partitions

2020 ◽  
Vol 70 (1) ◽  
pp. 29-40
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Set Partitions ◽  
Minimal Elements ◽  
Growth Functions ◽  
Asymptotic Formulae ◽  
Canonical Order ◽  
First Occurrence ◽  
Canonical Representations ◽  
The Mean

AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

Sign in / Sign up

Export Citation Format

Share Document