scholarly journals Wilf Classes of Pairs of Permutations of Length 4

10.37236/1922 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian Le
Keyword(s):  
Generating Function ◽  
Equivalence Classes ◽  
Order Relations ◽  
Set Of Permutations ◽  
Wilf Equivalence

$S_n(\pi_1,\pi_2,\dots, \pi_r)$ denotes the set of permutations of length $n$ that have no subsequence with the same order relations as any of the $\pi_i$. In this paper we show that $|S_n(1342,2143)|=|S_n(3142,2341)|$ and $|S_n(1342,3124)|=|S_n(1243,2134)|$. These two facts complete the classification of Wilf-equivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a "block", and in the former we find a generating function for $|S_n(1342,2143)|$.

scholarly journals On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

10.37236/2313 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Kendra Killpatrick
Keyword(s):  
Equivalence Classes ◽  
Catalan Numbers ◽  
Major Index ◽  
Powers Of 2 ◽  
Set Of Permutations ◽  
Adic Valuation ◽  
Permutation Statistic ◽  
Wilf Equivalence

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan.  In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$.  The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$.  We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result.  Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index.    In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$.  We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$.  In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

scholarly journals Wilf-Equivalence on $k$-ary Words, Compositions, and Parking Functions

10.37236/147 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Parking Functions ◽  
Arbitrary Size ◽  
Wilf Equivalence ◽  
Full Classification

In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.

scholarly journals Encoding and classification of permutations bу special conversion with estimates of class power

2019 ◽  
pp. 36-43
Author(s):  
M. Savchuk ◽  
M. Burlaka
Keyword(s):  
Equivalence Classes ◽  
Statistical Criterion ◽  
Random Permutations ◽  
High Quality ◽  
Interval Estimates ◽  
S Transform ◽  
Set Of Permutations ◽  
Class Power ◽  
Python Programming

Scientific articles investigating properties and estimates of the number of so-called complete permutations are surveyed and analyzed. The paper introduces a special S-transform on the set of permutations and determines the permutation properties according to this transform. Classification and coding of permutations by equivalence classes according to their properties with respect to S-transformation is proposed. This classification and permutation properties, in particular, generalize known results for complete permutations regarding determining certain cryptographic properties of substitutions that affect the cryptographic transformations security. The exact values of the number of permutations in equivalence classes for certain permutation sizes are calculated and the estimates of the cardinality of classes with various properties are constructed by statistical modeling. The complete list of permutation classes with the exact values of their sizes for permutations of order n = 11 is presented. The interval estimates for the size of classes with various characteristics for permutations of order n = 11, 26, 30, 31, 32, 33, 45, 55 are obtained. Monte Carlo estimates and bounds of confidence intervals used the approximation of the binomial distribution by the normal and Poisson distributions, as well as the Python programming language package Scipy. Statistical tables have been calculated that can be used for further conclusions and estimates. The classification of permutations by their properties with respect to the introduced transform can be used in constructing high-quality cryptographic transformations and transformations with special features. The classes of complete permutations with their properties are selected as the best for rotary cryptosystems applications. The obtained results can be used, in particular, to search for permutations with certain characteristics and properties, to find the probability that the characteristic of the generated permutation belongs to a collection of given characteristics, to estimate the complexity of finding permutations with certain properties. A statistical criterion of consent, which uses the characteristics of permutations by S-transformation to test the generators of random permutations and substitutions is proposed.

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

scholarly journals Classification of fused links

2016 ◽  
Vol 25 (14) ◽  
pp. 1650076 ◽  
Author(s):  
Timur Nasybullov
Keyword(s):  
Symmetric Group ◽  
Equivalence Classes ◽  
One To One ◽  
Text Component

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of [Formula: see text]-component fused links is in one-to-one correspondence with the set of elements of the abelization [Formula: see text] up to conjugation by elements from the symmetric group [Formula: see text].

scholarly journals A classification of intersection type systems

2002 ◽  
Vol 67 (1) ◽  
pp. 353-368
Author(s):  
M. W. Bunder
Keyword(s):  
Type Systems ◽  
Equivalence Classes ◽  
Type Assignment ◽  
Emptiness Problem ◽  
Intersection Types

AbstractThe first system of intersection types. Coppo and Dezani [3], extended simple types to include intersections and added intersection introduction and elimination rules ((ΛI ) and (ΛE) ) to the type assignment system. The major advantage of these new types was that they were invariant under β-equality, later work by Barendregt, Coppo and Dezani [1], extended this to include an (η) rule which gave types invariant under βη-reduction.Urzyczyn proved in [6] that for both these systems it is undecidable whether a given intersection type is empty. Kurata and Takahashi however have shown in [5] that this emptiness problem is decidable for the sytem including (η). but without (ΛI).The aim of this paper is to classify intersection type systems lacking some of (ΛI), (ΛE) and (η), into equivalence classes according to their strength in typing λ-terms and also according to their strength in possessing inhabitants.This classification is used in a later paper to extend the above (un)decidability results to two of the five inhabitation-equivalence classes. This later paper also shows that the systems in two more of these classes have decidable inhabitation problems and develops algorithms to find such inhabitants.

Generating functions for finite group actions on surfaces

1998 ◽  
Vol 124 (1) ◽  
pp. 21-49 ◽  
Author(s):  
C. MACLACHLAN ◽  
A. MILLER
Keyword(s):  
Finite Group ◽  
Generating Function ◽  
Generating Functions ◽  
Isomorphism Class ◽  
Group Actions ◽  
Equivalence Classes ◽  
Topological Conjugacy ◽  
Finite Group Actions ◽  
Orientable Surfaces

For a fixed finite group G, the numbers Ng of equivalence classes of orientation-preserving actions of G on closed orientable surfaces Σg of genus g can be encoded by a generating function [sum ]Ngzg. When equivalence is determined by the isomorphism class of the quotient orbifold Σg/G, we show that the generating function is rational. When equivalence is topological conjugacy, we examine the cases where G is abelian and show that the generating function is again rational in the cases where G is cyclic.

scholarly journals A Refined Count of Coxeter Element Reflection Factorizations

10.37236/7362 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Elise DelMas ◽  
Thomas Hameister ◽  
Victor Reiner
Keyword(s):  
Generating Function ◽  
Coxeter Element ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Coxeter Number ◽  
Simple Product

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the numberof reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

Sign in / Sign up

Export Citation Format

Share Document