The enumeration of generalized Tamari intervals

Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Pr\'eville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case $v=(NE)^n$. Our main contribution is that the enumeration of intervals in $\rm{T{\scriptsize AM}}(v)$, over all $v$ of length $n$, is given by $\frac{2 (3n+3)!}{(n+2)! (2n+3)!}$. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in $\rm{T{\scriptsize AM}}(v)$ to non-separable planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From generalized Tamari intervals to non-separable planar maps (extended abstract)", submitte

An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.Comment: 18 page

Self Presentation of Select Female Athletes on Instagram and Public Responses

Social media sites vastly have changed the ways in which athletes can interact with their fans. Positioned within the area of feminist cultural studies, this thesis employs a mixed-methods approach to examine how 8 selected female athletes display their bodies on Instagram via posted photos; how the public responds to this content via photo comments; and how these interactive postings fit with wider socio-cultural and historical contexts of women’s bodies in sport and physical activity. Using Erving Goffman’s theory on self-presentation and Jurgen Habermas’ theory on the public sphere, I conceive Instagram as both stage and space for public deliberation. Despite the autonomy over curating content that Instagram allows, images of women’s bodies were often perceived from a sexual lens regardless of the presence of sexualization in an image. Photo comments provide insight into public understanding of discourses on women’s bodies in sport, health and fitness, wherein sportswomen continued to be situated within traditional gender stereotypes. Four overarching themes were developed based on photo comments: 1) women as bodies first, and athletes second; 2) women as “women” first and athletes second; 3) the paradoxes between cultural constructions of femininity and athlete, with particular focus on the difference in treatment between black and white athletes and; 4) impression management and its role in reinforcing health and fitness as a consumable product

Defected and loyal? A case study of counter-defection mechanisms inside Chechen paramilitaries

Can former insurgents in the service of counterinsurgent paramilitaries be considered a perfectly loyal force? What mechanisms may help to deter subsequent defections of individuals who have already “betrayed” once? Drawing on a unique set of primary data, this article examines the effective counter-defection practices of Chechnya’s pro-Moscow paramilitaries toward prospective defectors from among ex-insurgents. It explores three interwoven mechanisms employed with various intensities to avert “double defections” at the peak of the locally fought counterinsurgency in Chechnya from 2000 to 2005. These mechanisms are: a) extrajudicial executions of recidivists and their relatives, b) initiation violence targeting insurgents’ relatives, and c) disclosure of the identities of defected insurgents who were responsible for killing insurgents in combat to the families of slain insurgents

Relationships Among A Football Specific Test, Wonderlic Test And Decision-Making In NCAA Football Players

Aim. This study compared two different evaluation methods of cognitive ability (Wonderlic and a Football Test) and their relationships to Assign Grade (accuracy of their decision making on the football field). Methods. Thirty-seven NCAA football players were given both tests, separated by several weeks. Additionally, the football coaches graded the players’ decision making performance with an “Assign Grade,” an evaluation of the players’ ability to make the correct decisions on the field. Results. Spearman rho correlation coefficient showed no significant correlations between the Wonderlic Personnel Test and Assign Grade (r = 0.04, p = 0.82) for any of the groups or player positions. Correlations were higher for the Football Test and Assign Grade (r = 0.30, p = 0.07), and a positive result was found for the Football Test and Assign Grade (r = 0.46, p = 0.04), when the specific positions were categorized as “more cognitive demanding.” Conclusion. There should be more research done to understand the cognitive demands of playing football and type of tests best predict performance

The representation of the symmetric group on m-Tamari intervals

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journa

The North Caucasus Insurgency: a Potential Spillover into the Russian Federation?

This contribution investigates the causes behind the downfall of the North Caucasus insurgency focusing on the pre-Sochi counter-insurgency and the massive outflow of foreign fighters to Syria and Iraq. In order to do so, this article provides a short historical background on the North Caucasus insurgency since the establishment of the Caucasus Emirate and discusses the threat of foreign fighters returning to the North Caucasus. It argues that although the insurgency has been crushed and entered a latent and incipient stage in 2014, the potential return of foreign fighters coupled with the recurrent discriminatory policies against Islam in Russia and the Islamic State's online propaganda might trigger a new spillover and upsurge of insurgent violence outside of the North Caucasus

The North Caucasus, the Future of Russia, and Foreign Fighters in Ukraine

The article documents the different groups of Caucasian foreign fighters in Ukraine and their relationships with other military groups. It then investigates the potential impact of those foreign fighters on the North Caucasus and the stability of the Russian Federation. It concludes by challenging the assumption that the North Caucasus might become the epicenter of a national movement leading to the collapse of Russia and suggesting that insurgent activities might increase in Western Russia instead

Optimal learning in optimal contexts : the role of self-determination in education.

This literature review provides an overview of education studies that have been guided by selfdetermination theory (SDT). First, the authors examine studies that have assessed motivation based on SDT. Second, the authors detail research that has focussed on the linkages between motivation types and students’ behavioural, affective, and cognitive outcomes. Third, the authors present studies on how learning contexts (parents, teachers) contribute to students’ motivational resources. The authors conclude that the motivation types proposed by SDT are important for understanding how students thrive and succeed at school. The authors also highlight the significant role played by teachers and parents in the development of student motivation. The authors conclude with a summary of the benefits of selfmotivation for learning and offer some recommendations for the field.Le présent article procure une vue d'ensemble des études sur l'éducation qui ont été guidées par la théorie de l'autodétermination. En premier lieu, nous examinons les études qui ont évalué la motivation fondée sur la théorie de l'autodétermination. Nous expliquons ensuite la recherche portant sur les liens existant entre les types de motivation et les résultats des étudiants en matière de comportement et de composantes affectives et cognitives. Nous présentons aussi les études sur les contextes d'apprentissage (parents, enseignants) qui contribuent aux ressources motivationnelles des êtudiants. Nous en déduisons que les types de motivation proposés par la théorie de l'autodétermination permettent de comprendre les éléments qui permettent aux étudiants de se démarquer et de réussir en milieu scolaire. Nous mettons aussi en relief le rôle important que jouent les enseignants et les parents dans le développement de la motivation de l'éudiant. En conclusion, nous présentons un résumé des bienfaits de l'automotivation dans le cadre de l'apprentissage et offrons quelques recommandations pertinentes au domaine en question