scholarly journals Refining Enumeration Schemes to Count According to Permutation Statistics

10.37236/4002 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Andrew M. Baxter
Keyword(s):  
Permutation Statistics ◽  
Major Index ◽  
Enumeration Schemes ◽  
Pattern Avoiding Permutations ◽  
Enumeration Scheme

We develop algorithmic tools to compute quickly the distribution of permutation statistics over sets of pattern-avoiding permutations. More specfically, the algorithms are based on enumeration schemes, the permutation statistics are based on the number of occurrences of certain vincular patterns, and the permutations avoid sets of vincular patterns. We prove that whenever a finite enumeration scheme exists to count the number of pattern-avoiding permutations, then the distribution of statistics like the number of descents can also be computed based on the same scheme. Statistics such as the number of peaks, right-to-left maxima, and the major index are also investigated, as well as multi-statistics.

scholarly journals Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

10.37236/2001 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Petter Brändén ◽  
Anders Claesson
Keyword(s):  
Fixed Points ◽  
Relative Position ◽  
Permutation Statistics ◽  
Permutation Patterns ◽  
Alternating Permutations ◽  
Major Index ◽  
Permutation Pattern ◽  
New Type ◽  
Permutation Statistic ◽  
Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

scholarly journals Recursions and divisibility properties for combinatorial Macdonald polynomials

2011 ◽  
Vol Vol. 13 no. 1 (Combinatorics) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Elizabeth Niese
Keyword(s):  
Hilbert Series ◽  
Permutation Statistics ◽  
Macdonald Polynomial ◽  
Integer Partition ◽  
Theoretical Methods ◽  
Major Index ◽  
International Audience ◽  
Definition Of ◽  
Fermionic Formulas ◽  
Divisibility Properties

Combinatorics International audience For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

scholarly journals Graphical Mahonian Statistics on Words

10.37236/6263 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Amy Grady ◽  
Svetlana Poznanović
Keyword(s):  
Directed Graphs ◽  
Permutation Statistics ◽  
Major Index ◽  
Mahonian Statistics

Foata and Zeilberger defined the graphical major index, $\mathrm{maj}_U$, and the graphical inversion index, $\mathrm{inv}_U$, for words over the alphabet $\{1, 2, \dots, n\}$. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and $\mathrm{inv}$ indexed by directed graphs $U$. They showed that $\mathrm{maj}_U$ and $\mathrm{inv}_U$ are equidistributed over all rearrangement classes if and only if $U$ is bipartitional. In this paper we strengthen their result by showing that if $\mathrm{maj}_U$ and $\mathrm{inv}_U$ are equidistributed on a single rearrangement class then $U$ is essentially bipartitional. Moreover, we define a graphical sorting index, $\mathrm{sor}_U$, which generalizes the sorting index of a permutation. We then characterize the graphs $U$ for which $\mathrm{sor}_U$ is equidistributed with $\mathrm{inv}_U$ and $\mathrm{maj}_U$ on a single rearrangement class. 

scholarly journals Inversion Generating Functions for Signed Pattern Avoiding Permutations

10.37236/6545 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Naiomi T. Cameron ◽  
Kendra Killpatrick
Keyword(s):  
Generating Functions ◽  
Combinatorial Proof ◽  
Hyperoctahedral Group ◽  
Major Index ◽  
Signed Permutations ◽  
Mahonian Statistics ◽  
Almost All ◽  
Open Question ◽  
Pattern Avoiding Permutations

We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two.  In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$.  We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$.  In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$.  The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$

Implicit Generation of Pattern-Avoiding Permutations by Using Permutation Decision Diagrams

2014 ◽  
Vol E97.A (6) ◽  
pp. 1171-1179
Author(s):  
Yuma INOUE ◽  
Takahisa TODA ◽  
Shin-ichi MINATO
Keyword(s):  
Decision Diagrams ◽  
Pattern Avoiding Permutations

Characteristics analysis of behavioural portfolio theory in the Markowitz portfolio theory framework

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Saksham Mittal ◽  
Sujoy Bhattacharya ◽  
Satrajit Mandal
Keyword(s):  
Emerging Markets ◽  
Asset Allocation ◽  
Portfolio Theory ◽  
Optimal Portfolio ◽  
Content Type ◽  
Major Index ◽  
Brics Countries ◽  
Trade Offs ◽  
Characteristics Analysis ◽  
Mean Variance

PurposeIn recent times, behavioural models for asset allocation have been getting more attention due to their probabilistic modelling for scenario consideration. Many investors are thinking about the trade-offs and benefits of using behavioural models over conventional mean-variance models. In this study, the authors compare asset allocations generated by the behavioural portfolio theory (BPT) developed by Shefrin and Statman (2000) against the Markowitz (1952) mean-variance theory (MVT).Design/methodology/approachThe data used have been culled from BRICS countries' major index constituents from 2009 to 2019. The authors consider a single period economy and generate future probable outcomes based on historical data in order to determine BPT optimal portfolios.FindingsThis study shows that a fair number of portfolios satisfy the first entry constraint of the BPT model. BPT optimal portfolio exhibits high risk and higher returns as compared to typical Markowitz optimal portfolio.Originality/valueThe BRICS countries' data were used because the dynamics of the emerging markets are significantly different from the developed markets, and many investors have been considering emerging markets as their new investment avenues.

scholarly journals Las Elecciones Olvidadas: Reflexiones En Torno A La Participación Ciudadana Locales En El Estado De Hidalgo (1993-2005)

Xihmai ◽  
2012 ◽  
Vol 2 (4) ◽  
Author(s):  
Enrique López Rivera
Keyword(s):  
The State ◽  
Major Index ◽  
Present Text

Resumen Las elecciones de diputados locales son las que concentran el mayor í­ndice de abstencionismo en el estado de Hidalgo. Ese comportamiento puede ser explicado a través de algunos factores de corte sociodemográfico que ayudan a comprender los contrastes y contradicciones de la conducta del votante y del abstencionista. ¿Cuáles de factores ayudan a entender con mayor precisión este fenómeno? Esa es la pregunta que trata de contestar el presente texto. Abstract The elections of local deputies are those that concentrate the major index of abstentionism in the state of Hidalgo. This behavior can be made clear across some factors of court sociodemográfico that they help to understand the contrasts and contradictions of the conduct of the voter and of the abstentionist. Which of factors do help to understand with major precision this phenomenon? This it is the question that tries to answer the present text.

scholarly journals SEPTEM VERBA A CHRISTO W STAROPOLSKICH APOKRYFACH NOWOTESTAMENTALNYCH W PERSPEKTYWIE ŹRÓDŁOZNAWCZEJ I LINGWISTYCZNEJ. CZĘŚĆ DRUGA: JĘZYKOWE UKSZTAŁTOWANIE

2020 ◽  
pp. 159-170
Author(s):  
OLGA ZIÓŁKOWSKA
Keyword(s):  
Regular Word ◽  
Religious Instruction ◽  
Enumeration Schemes ◽  
The Cross ◽  
Modal Frame

This article is the second and the last part of a series dedicated to the characteristics of a description of Christ’s seven words uttered on the cross as accounted in Old Polish biblical and apocryphal narrations. In part one, I focus on the origin of the words, the characteristics of Old Polish texts containing them. I highlight the most important differences in the narration of the specific fragments of Old Polish Passions of Jesus. Part two is entirely dedicated to the language of the fragments of Old Polish texts on Christ’s seven words uttered on the cross (ŻPJK, SCh and RD). First, I present the enumeration schemes in each apocrypha. They are strictly related to the tradition of religious instruction and teaching and are an attempt at sorting out the material. The article presents also the various ways in which Christ’s specific words are called. The regular word-forming structure of the modifiers affects the rhythmic form of the specific fragments of texts. Finally, I show how quotations from Christ were introduced into each apocrypha: how the utterances’ modal frame was shaped and what verbs of speech were used. It turns out that in each historic text, Christ’s words were treated differently: in the SCh, they were described most extensively and in the RD – least extensively but it is the RD where the enumerations are most precise with respect to the syntax, perhaps because the specific parts are at the smallest distance from each other. Christ’s seven words on the cross are least structured in the ŻPJK.

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 MODEL FOR RAPID ASSESSMENT OF VULNERABILITY OF OFFICE BUILDINGS TO BLAST USING SWARA AND SMART METHODS (A CASE STUDY OF SWISS RE TOWER)

2016 ◽  
Vol 22 (6) ◽  
pp. 831-843 ◽  
Author(s):  
Jalal NAKHAEI ◽  
Mahdi BITARAFAN ◽  
Shahin LALE AREFI ◽  
Oleg KAPLIŃSKI
Keyword(s):  
Rapid Assessment ◽  
Office Buildings ◽  
Architectural Form ◽  
Major Index ◽  
World Demand ◽  
Medium Level ◽  
Four Levels ◽  
And Function ◽  
Building Form

Accidental and intentional explosions are incidents often destroying buildings and leaving casualties. As a re­sult of these blasts all over the world, demand of safe constructions with less vulnerability to explosions is rising. A large number of office buildings are built each year in many countries, housing large numbers of staff and clients, and due to specific nature and function, activities and services, these buildings are usually centrally located. Their architectural form being vital, therefore the article attempts, firstly, to present indices depicting the building form from the viewpoint of vulnerability to explosion. Secondly, the article presents such indexes as: capability to reduce blast effects, economic factors, simplicity of implementation, relationship among spaces in the crisis condition, and creating the least unusable space. The model of rapid assessment of vulnerability of office buildings forms to blast, SMART (simple multi attribute ranking technique) procedure is used and, applying the SWARA method, the weight of each major index and sub-index is arrived at. The model presented in the paper shows the assessment systems using figures between zero and a hundred, and four levels of vulnerability: weak, medium, good and excellent. The closer the figure to a hundred, the lower the vulnerability of the office building forms to blast. Swiss Re Tower case study was presented in the article rating of vul­nerability of this building against explosion. It was found to be equal to 62.11%, and its standing was at medium level.

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