scholarly journals Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture

2020 ◽  
Vol 8 ◽  
Author(s):  
Rebecca Patrias ◽  
Oliver Pechenik
Keyword(s):  
Algebraic Combinatorics ◽  
Plane Partition ◽  
Full Generality ◽  
Minimal Elements ◽  
Plane Partitions

Abstract One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995): consider a plane partition P in an $a \times b \times c$ box ${\sf B}$ . Let $\Psi (P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$ . Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the $\Psi $ -orbit of P is always a multiple of p. This conjecture was established for $p \gg 0$ by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of p in work of K. Dilks, J. Striker and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.

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 On the Zeros of Plane Partition Polynomials

10.37236/2026 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Robert P. Boyer ◽  
Daniel T. Parry
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Plane Partition ◽  
Limiting Behavior ◽  
Plane Partitions ◽  
Scale Behavior

Let $PL(n)$ be the number of all plane partitions of $n$ while $pp_k(n)$ be the number of plane partitions of $n$ whose trace is exactly $k$. We study the zeros of polynomial versions $Q_n(x)$ of plane partitions where $Q_n(x) = \sum pp_k(n) x^k$. Based on the asymptotics we have developed for $Q_n(x)$ and computational evidence, we determine the limiting behavior of the zeros of $Q_n(x)$ as $n\to\infty$. The distribution of the zeros has a two-scale behavior which has order $n^{2/3}$ inside the unit disk while has order $n$ on the unit circle.

scholarly journals Plane partitions of shifted double staircase shape

2021 ◽  
Vol 183 ◽  
pp. 105486
Author(s):  
Sam Hopkins ◽  
Tri Lai
Keyword(s):  
Plane Partitions

scholarly journals Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 728
Author(s):  
Yasunori Maekawa ◽  
Yoshihiro Ueda
Keyword(s):  
Hyperbolic System ◽  
Dissipative Structure ◽  
Dissipative Structures ◽  
Algebraic Characterization ◽  
Symmetric Hyperbolic System ◽  
Full Generality ◽  
Order Linear ◽  
First Order

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

scholarly journals On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

scholarly journals Dynamic Shortest Paths Methods for the Time-Dependent TSP

Algorithms ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 21
Author(s):  
Christoph Hansknecht ◽  
Imke Joormann ◽  
Sebastian Stiller
Keyword(s):  
Column Generation ◽  
Traveling Salesman Problem ◽  
Shortest Path ◽  
Valid Inequalities ◽  
Shortest Paths ◽  
Traveling Salesman ◽  
Time Dependent ◽  
Full Generality ◽  
Branching Rule ◽  
The Traveling Salesman Problem

The time-dependent traveling salesman problem (TDTSP) asks for a shortest Hamiltonian tour in a directed graph where (asymmetric) arc-costs depend on the time the arc is entered. With traffic data abundantly available, methods to optimize routes with respect to time-dependent travel times are widely desired. This holds in particular for the traveling salesman problem, which is a corner stone of logistic planning. In this paper, we devise column-generation-based IP methods to solve the TDTSP in full generality, both for arc- and path-based formulations. The algorithmic key is a time-dependent shortest path problem, which arises from the pricing problem of the column generation and is of independent interest—namely, to find paths in a time-expanded graph that are acyclic in the underlying (non-expanded) graph. As this problem is computationally too costly, we price over the set of paths that contain no cycles of length k. In addition, we devise—tailored for the TDTSP—several families of valid inequalities, primal heuristics, a propagation method, and a branching rule. Combining these with the time-dependent shortest path pricing we provide—to our knowledge—the first elaborate method to solve the TDTSP in general and with fully general time-dependence. We also provide for results on complexity and approximability of the TDTSP. In computational experiments on randomly generated instances, we are able to solve the large majority of small instances (20 nodes) to optimality, while closing about two thirds of the remaining gap of the large instances (40 nodes) after one hour of computation.

scholarly journals Role-reference associations and the explanation of argument coding splits

Linguistics ◽  
2020 ◽  
Vol 59 (1) ◽  
pp. 123-174
Author(s):  
Martin Haspelmath
Keyword(s):  
Language Use ◽  
General Pattern ◽  
General Level ◽  
Full Generality ◽  
Coding Efficiency ◽  
Object Marking ◽  
Case Markers ◽  
Ditransitive Constructions

Abstract Argument coding splits such as differential (= split) object marking and split ergative marking have long been known to be universal tendencies, but the generalizations have not been formulated in their full generality before. In particular, ditransitive constructions have rarely been taken into account, and scenario splits have often been treated separately. Here I argue that all these patterns can be understood in terms of the usual association of role rank (highly ranked A and R, low-ranked P and T) and referential prominence (locuphoric person, animacy, definiteness, etc.). At the most general level, the role-reference association universal says that deviations from usual associations of role rank and referential prominence tend to be coded by longer grammatical forms. In other words, A and R tend to be referentially prominent in language use, while P and T are less prominent, and when less usual associations need to be expressed, languages often require special coding by means of additional flags (case-markers and adpositions) or additional verbal voice coding (e.g., inverse or passive markers). I argue that role-reference associations are an instance of the even more general pattern of form-frequency correspondences, and that the resulting coding asymmetries can all be explained by frequency-based predictability and coding efficiency.

Current research on algebraic combinatorics

1986 ◽  
Vol 2 (1) ◽  
pp. 287-308 ◽  
Author(s):  
Eiichi Bannai ◽  
Tatsuro Ito
Keyword(s):  
Algebraic Combinatorics
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