Richard Stanley through a crystal lens and from a random angle

We review Stanley's seminal work on the number of reduced words of the longest element of the symmetric group and his Stanley symmetric functions. We shed new light on this by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion. We also view this from a random perspective and study a Markov chain on reduced words of the longest element in a finite Coxeter group, in particular the symmetric group, and mention a generalization to a poset setting.Comment: 11 pages; 3 figures; v2 updated references and added discussion on Coxeter-Knuth grap

Orbits of Plane Partitions of Exceptional Lie Type

For each minuscule flag variety $X$, there is a corresponding minuscule poset, describing its Schubert decomposition. We study an action on plane partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass (1995). For plane partitions of height at most $2$, D. Rush and X. Shi (2013) proved an instance of the cyclic sieving phenomenon, completely describing the orbit structure of this action. They noted their result does not extend to greater heights in general; however, when $X$ is one of the two minuscule flag varieties of exceptional Lie type $E$, they conjectured explicit instances of cyclic sieving for all heights. We prove their conjecture in the case that $X$ is the Cayley-Moufang plane of type $E_6$. For the other exceptional minuscule flag variety, the Freudenthal variety of type $E_7$, we establish their conjecture for heights at most $4$, but show that it fails generally. We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case $X$ is an even-dimensional quadric hypersurface. Our argument uses ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on plane partitions to combinatorics derived from $K$-theoretic Schubert calculus.Comment: 25 pages, 7 figures, 3 tables. Section 5 rewritten and simplifie

Le livre dans les couvents mendiants à la fin de l’Ancien Régime, d’après l’enquête nationale de 1790-1791

International audienceThe seizing of clerical libraries between 1789 and 1791 accounted for the implementation of a novel cultural policy which itself led to the creation of the first public libraries in 1803. The rules set by the revolutionary government and then implemented by the city councils and the districts are based on a survey on the number and the quality of the seized books. For all its flaws that survey brings forth the picture of what church culture was like in the late 18th century.As far as mendicant orders are concerned, that picture highlights the importance of urban covents with the finest libraries being situated in the national, provincial and episcopal capitals whereas rural convents enjoy lesser intellectual ressources. Such imbalance is to be related to the age old convent establishments, the existence of institutions for the training of friars in city convents and a thriving cultural environment from which the supporters and the benefactors originated. These outstanding links between the mendicants and the cities enabled these clerics to have prominence over the regular orders.Entre 1789 et 1791, la saisie des bibliothèques du clergé engendre en France la mise en place d’une politique culturelle nouvelle, qui aboutira, après bien des tâtonnements, à la création des premières bibliothèques publiques en 1803. La législation produite par les autorités révolutionnaires, puis mise en œuvre dans les municipalités et les districts, est motivée par une enquête sur la quantité et la qualité des livres mis sous séquestre. Cette enquête, malgré ses imperfections, brosse le tableau de la culture ecclésiastique à la fin du XVIIIe siècle. À l’échelle des ordres mendiants, ce tableau met en évidence l’importance de l’implantation urbaine des couvents, les plus belles bibliothèques se trouvant dans les capitales (nationale, provinciales et épiscopales), tandis que les couvents des champs ont des ressources intellectuelles moindres. Ces déséquilibres sont à mettre en lien avec l’ancienneté des fondations conventuelles, la présence des institutions de formation des profès dans les couvents urbains, la présence d’un milieu culturel dynamique d’où sont issus les protecteurs et bienfaiteurs des ordres mendiants. Ces liens culturels privilégiés entre les mendiants et les villes singularisent fortement ces religieux par rapport aux autres ordres réguliers

Deduction modulo theory

This paper is a survey on Deduction modulo theor

Piecewise-linear and birational toggling

We define piecewise-linear and birational analogues of the toggle-involutions on order ideals of posets studied by Striker and Williams and use them to define corresponding analogues of rowmotion and promotion that share many of the properties of combinatorial rowmotion and promotion. Piecewise-linear rowmotion (like birational rowmotion) admits an alternative definition related to Stanley's transfer map for the order polytope; piecewise-linear promotion relates to Sch\"utzenberger promotion for semistandard Young tableaux. The three settings for these dynamical systems (combinatorial, piecewise-linear, and birational) are intimately related: the piecewise-linear operations arise as tropicalizations of the birational operations, and the combinatorial operations arise as restrictions of the piecewise-linear operations to the vertex-set of the order polytope. In the case where the poset is of the form $[a] \times [b]$, we exploit a reciprocal symmetry property recently proved by Grinberg and Roby to show that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. This yields a new proof of a theorem of Cameron and Fon-der-Flaass. Our proofs make use of the correspondence between rowmotion and promotion orbits discovered by Striker and Williams, which we make more concrete. We also prove some homomesy results, showing that for certain functions $f$, the average value of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen.Comment: This is essentially a synopsis of the longer article-in-progress arXiv:1310.5294 "Combinatorial, piecewise-linear, and birational homomesy for products of two chains" by David Einstein and James Propp. It was prepared for FPSAC 2014, and will appear along with the other FPSAC 2014 extended abstracts in a special issue of the journal Discrete Mathematics and Theoretical Computer Scienc

Deciding First-Order Satisfiability when Universal and Existential Variables are Separated

We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones -- the Bernays-Sch\"onfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property and derive a non-elementary upper bound on the computing time required for deciding satisfiability in the new fragment. For the subfragment of prenex sentences with the quantifier prefix $\exists^* \forall^* \exists^*$ the satisfiability problem is shown to be complete for NEXPTIME. Finally, we discuss how automated reasoning procedures can take advantage of our results.Comment: Extended version of our LICS 2016 conference paper, 23 page