scholarly journals The $q=-1$ phenomenon for bounded (plane) partitions via homology concentration

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
P. Hersh ◽  
J. Shareshian ◽  
D. Stanton
Keyword(s):  
Fixed Points ◽  
General Framework ◽  
Plane Partitions ◽  
International Audience

International audience Algebraic complexes whose "faces'' are indexed by partitions and plane partitions are introduced, and their homology is proven to be concentrated in even dimensions with homology basis indexed by fixed points of an involution, thereby explaining topologically two quite important instances of Stembridge's $q=-1$ phenomenon. A more general framework of invariant and coinvariant complexes with coefficients taken $\mod 2$ is developed, and as a part of this story an analogous topological result for necklaces is conjectured. Complexes algébriques dont les "faces'' sont indexées par des partitions et des partitions planes sont introduits. Il est démontré que leur homologie est concentrée en dimensions paires, avec base de homologie indexée par des points fixes d'une involution. Ce résultat explique d'une manière topologique deux instances du phénomène $q=-1$ dû a Stembridge. De plus, un cadre plus général des complexes invariants et coinvariants dont les coefficients sont pris modulo $2$ est développé. Comme part de cette histoire, nous conjecturons un résultat analogue pour des colliers.

scholarly journals The Gaussian free field and strict plane partitions

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Mirjana Vuletić
Keyword(s):  
Free Field ◽  
Descent Method ◽  
Dirichlet Boundary Conditions ◽  
Steepest Descent Method ◽  
Gaussian Free Field ◽  
Correlation Kernel ◽  
Plane Partitions ◽  
International Audience ◽  
Pfaffian Formula ◽  
The Laplace Operator

International audience We study height fluctuations around the limit shape of a measure on strict plane partitions. It was shown in our earlier work that this measure is a Pfaffian process. We show that the height fluctuations converge to a pullback of the Green's function for the Laplace operator with Dirichlet boundary conditions on the first quadrant. We use a Pfaffian formula for higher moments to show that the height fluctuations are governed by the Gaussian free field. The results follow from the correlation kernel asymptotics which is obtained by the steepest descent method.

scholarly journals Enumeration of Cylindric Plane Partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Robin Langer
Keyword(s):  
Natural Generalization ◽  
Path Model ◽  
Plane Partition ◽  
Combinatorial Interpretation ◽  
Plane Partitions ◽  
Generating Series ◽  
International Audience ◽  
The Individual ◽  
The Right ◽  
Reverse Plane

International audience Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions. Les partitions planes cylindriques sont une généralisation naturelle des partitions planes renversées. Une série génératrice pour énumération des partitions planes cylindriques a été donnée récemment par Borodin. Comme dans le cas des partitions planes renversées, la partie droite de cette identité peut être factoriser en terme de "longueur d’équerres'' des carrés dans la forme sous-jacente. Le premier résultat de cet article est une nouvelle preuve bijective de l'identité de Borodin qui utilise le cadre de "diagramme de croissance'' de Fomin pour la correspondance de RSK généralisée. Le deuxième résultat de cette article est une $(q,t)$-déformation d'identité de Borodin qui généralise un résultat de Okada dans le cas des partitions planes renversées. Le troisième résultat de cet article est une formule combinatoire explicite pour le poids de Macdonald qui utilise le modèle des chemins non-intersectant pour les partitions planes cylindriques.

scholarly journals The multivariate arithmetic Tutte polynomial

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Petter Brändèn ◽  
Luca Moci
Keyword(s):  
General Framework ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Nous Proposons ◽  
International Audience

International audience We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial. Nous introduisons une version arithmétique du polynôme de Tutte multivariée récemment étudié par Sokal, et un quasi-polynôme qui interpole entre les deux. Nous proposons une représentation de Fortuin-Kasteleyn neutralise pour les matroïdes arithmétiques représentables, avec des applications aux colorations et flux arithmétiques. Nous donnons une nouvelle preuve de la positivité des coefficients du polynôme de Tutte arithmétique dans le cadre plus général des matroïdes pseudo-arithmétiques. Dans le cas d'un matroïde arithmétique représentable, nous proposons une interprétation géométrique des coefficients du polynôme de Tutte arithmétique.

scholarly journals Plane partitions and the combinatorics of some families of reduced Kronecker coefficients.

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Laura Colmenarejo
Keyword(s):  
Generating Function ◽  
Plane Partitions ◽  
Rate Of Growth ◽  
International Audience ◽  
Kronecker Coefficients

International audience We compute the generating function of some families of reduced Kronecker coefficients. We give a combi- natorial interpretation for these coefficients in terms of plane partitions. This unexpected relation allows us to check that the saturation hypothesis holds for the reduced Kronecker coefficients of our families. We also compute the quasipolynomial that govern these families, specifying the degree and period. Moving to the setting of Kronecker co- efficients, these results imply some observations related to the rate of growth experienced by the families of Kronecker coefficients associated to the reduced Kronecker coefficients already studied.

scholarly journals Hadamard matrices of order 36 and double-even self-dual [72,36,12] codes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Iliya Bouyukliev ◽  
Veerle Fack ◽  
Joost Winne
Keyword(s):  
Fixed Points ◽  
Hadamard Matrices ◽  
Dual Codes ◽  
International Audience

International audience Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known. We found 7238 Hadamard matrices of order 36 and 522 inequivalent [72,36,12] double-even self-dual codes which are obtained from all 2-(35,17,8) designs with an automorphism of order 3 and 2 fixed points and blocks.

scholarly journals Analysis of Digital Expansions of Minimal Weight

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Florian Heigl ◽  
Clemens Heuberger
Keyword(s):  
Dynamic Programming ◽  
General Framework ◽  
Programming Approach ◽  
Random Input ◽  
Dynamic Programming Approach ◽  
Minimal Weight ◽  
Elliptic Curve Cryptosystems ◽  
Frobenius Endomorphism ◽  
International Audience ◽  
Digit Expansions

International audience Extending an idea of Suppakitpaisarn, Edahiro and Imai, a dynamic programming approach for computing digital expansions of minimal weight is transformed into an asymptotic analysis of minimal weight expansions. The minimal weight of an optimal expansion of a random input of length $\ell$ is shown to be asymptotically normally distributed under suitable conditions. After discussing the general framework, we focus on expansions to the base of $\tau$, where $\tau$ is a root of the polynomial $X^2- \mu X + 2$ for $\mu \in \{ \pm 1\}$. As the Frobenius endomorphism on a binary Koblitz curve fulfils the same equation, digit expansions to the base of this $\tau$ can be used for scalar multiplication and linear combination in elliptic curve cryptosystems over these curves.

scholarly journals A new generation tree for permutations, preserving the number of fixed points

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Philippe Duchon ◽  
Romaric Duvignau
Keyword(s):  
Fixed Points ◽  
Random Number ◽  
Random Number Generator ◽  
Specific Property ◽  
Number Generator ◽  
Random Generation ◽  
Combinatorial Algorithm ◽  
International Audience ◽  
Uniform Generation ◽  
New Generation

International audience We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.

scholarly journals Cyclic Sieving for Plane Partitions and Symmetry

2020 ◽  
Author(s):  
Sam Hopkins ◽  
Keyword(s):  
Fixed Points ◽  
Generating Function ◽  
Cyclic Group ◽  
Product Formula ◽  
Roots Of Unity ◽  
Plane Partitions ◽  
Cyclic Sieving Phenomenon ◽  
Combinatorial Set ◽  
Symmetric Plane ◽  
Cyclic Sieving

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

scholarly journals Latin square Thue-Morse sequences are overlap-free

2007 ◽  
Vol Vol. 9 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
C. Robinson Tompkins
Keyword(s):  
Fixed Points ◽  
Latin Square ◽  
International Audience

Automata, Logic and Semantics International audience We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche - Shallit and Frid.

scholarly journals On the link pattern distribution of quarter-turn symmetric FPL configurations

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Philippe Duchon
Keyword(s):  
Nous Obtenons ◽  
Plane Partitions ◽  
New Class ◽  
International Audience ◽  
Pattern Distribution ◽  
Special Case ◽  
Link Pattern

International audience We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions. Nous présentons de nouvelles conjectures portant sur la distribution des schémas de couplage des configurations de boucles compactes (FPL) invariantes, ou presque invariantes, par une rotation d'un quart de tour. Ces nouvelles conjectures étendent des conjectures précédentes dues à Razumov et Stroganov et à de Gier. Dans chaque cas, nous prouvons un cas particulier, en démontrant que le schéma de couplage conjecturé pour être le plus rare a effectivement la probabilité prédite. Nous obtenons également une formule pour l'énumération d'une nouvelle classe de quasi-symétrie de partitions planes.

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