scholarly journals Some Instances of Homomesy Among Ideals of Posets

10.37236/6051 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Shahrzad Haddadan
Keyword(s):  
Order Ideal ◽  
Combinatorial Objects ◽  
Order Ideals ◽  
Set Up

Given a permutation $\tau$ defined on a set of combinatorial objects $S$, together with some statistic $f:S\rightarrow \mathbb{R}$, we say that the triple $\langle S, \tau,f \rangle$ exhibits homomesy if $f$ has the same average along all orbits of $\tau$ in $S$. This phenomenon was observed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). Propp and Roby studied homomesy in the set of order ideals in the product of two chains, with two well known permutations, rowmotion and promotion, the statistic being the size of the order ideal. In this paper we extend their results to generalized rowmotion and promotion, together with a wider class of statistics in the product of two chains. Moreover, we derive similar results in other simply described posets. We believe that the framework we set up here can be fruitful in demonstrating homomesy results in order ideals of broader classes of posets. 

scholarly journals Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets

10.37236/4334 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Darij Grinberg ◽  
Tom Roby
Keyword(s):  
Finite Order ◽  
Disjoint Union ◽  
Parallel Analysis ◽  
Birational Map ◽  
Finite Poset ◽  
Order Ideals ◽  
Set Up ◽  
Grafting Onto

We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.

scholarly journals An Analogue of Covering Space Theory for Ranked Posets

10.37236/1576 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Michael E. Hoffman
Keyword(s):  
Ordered Set ◽  
Universal Cover ◽  
Order Ideal ◽  
Space Theory ◽  
Young Diagrams ◽  
Simple Description ◽  
Locally Finite ◽  
Combinatorial Objects ◽  
A Chain ◽  
Standard Young Tableaux

Suppose $P$ is a partially ordered set that is locally finite, has a least element, and admits a rank function. We call $P$ a weighted-relation poset if all the covering relations of $P$ are assigned a positive integer weight. We develop a theory of covering maps for weighted-relation posets, and in particular show that any weighted-relation poset $P$ has a universal cover $\tilde P\to P$, unique up to isomorphism, so that 1. $\tilde P\to P$ factors through any other covering map $P'\to P$; 2. every principal order ideal of $\tilde P$ is a chain; and 3. the weight assigned to each covering relation of $\tilde P$ is 1. If $P$ is a poset of "natural" combinatorial objects, the elements of its universal cover $\tilde P$ often have a simple description as well. For example, if $P$ is the poset of partitions ordered by inclusion of their Young diagrams, then the universal cover $\tilde P$ is the poset of standard Young tableaux; if $P$ is the poset of rooted trees ordered by inclusion, then $\tilde P$ consists of permutations. We discuss several other examples, including the posets of necklaces, bracket arrangements, and compositions.

scholarly journals Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup

2001 ◽  
Vol 28 (9) ◽  
pp. 535-543 ◽  
Author(s):  
Michiro Kondo
Keyword(s):  
Open Problem ◽  
Order Ideal ◽  
Adjoint Semigroup ◽  
Order Ideals ◽  
One To One ◽  
The Relationship

We consider the relationship between ideals of a BCI-algebra and order ideals of its adjoint semigroup. We show that (1) ifIis an ideal, thenI=M−1(M(I)), (2)M(M−1(J))is the order ideal generated byJ∩R(X), (3) ifXis a BCK-algebra, thenJ=M(M−1(J))for any order idealJofX, thus, for each BCK-algebraXthere is a one-to-one correspondence between the setℐ(X)of all ideals ofXand the set𝒪(X)of all order ideals of it, and (4) the orderM(M−1(J))is an order ideal if and only ifM−1(J)is an ideal. These results are the generalization of those denoted by Huang and Wang (1995) and Li (1999). We can answer the open problem of Li affirmatively.

scholarly journals The poset perspective on alternating sign matrices

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Symmetric Plane ◽  
International Audience ◽  
Order Ideals ◽  
Nous Utilisons

International audience Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

scholarly journals Birational Rowmotion and Coxeter-Motion on Minuscule Posets

10.37236/9557 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Soichi Okada
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Order Ideal ◽  
Positive Real ◽  
Finite Poset ◽  
Coxeter Number ◽  
Order Ideals ◽  
Minuscule Poset

Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistics. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.

scholarly journals GROUP EXTENSIONS AND THE PRIMITIVE IDEAL SPACES OF TOEPLITZ ALGEBRAS

2007 ◽  
Vol 49 (1) ◽  
pp. 81-92 ◽  
Author(s):  
SRIWULAN ADJI ◽  
IAIN RAEBURN ◽  
RIZKY ROSJANUARDI
Keyword(s):  
Structure Theorem ◽  
Group Extension ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Order Ideal ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Trivial Group ◽  
Order Ideals ◽  
The Ideal

Abstract.Let Γ be a totally ordered abelian group andIan order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebraT(Γ) to the structure of the Toeplitz algebrasT(I) andT(Γ/I). We then describe the primitive ideal space of the Toeplitz algebraT(Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure ofT(Γ) when 0→I→ Γ→ Γ/I→ 0 is a non-trivial group extension.

scholarly journals Invariant Principal Order Ideals under Foata’s Transformation

10.37236/2680 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Teresa X.S. Li ◽  
Melissa Y.F. Miao
Keyword(s):  
Symmetric Group ◽  
Bruhat Order ◽  
Weak Order ◽  
Invariance Property ◽  
Order Ideal ◽  
Principal Order ◽  
Fundamental Transformation ◽  
Order Ideals

Let $\Phi$ denote  Foata's second fundamental transformation on permutations. For a permutation $\sigma$ in the symmetric group $S_n$, let $\widetilde{\Lambda}_{\sigma}=\{\pi\in S_n\colon\pi\leq_{w} \sigma\}$ be the principal order ideal generated by $\sigma$  in the weak order $\leq_{w}$. Björner and Wachs have shown that $\widetilde{\Lambda}_{\sigma}$ is invariant under $\Phi$ if and only if $\sigma$ is a 132-avoiding permutation. In this paper, we consider the invariance property of  $\Phi$ on the principal order ideals ${\Lambda}_{\sigma}=\{\pi\in S_n\colon \pi\leq \sigma\}$ with respect to the Bruhat order $\leq$.  We obtain a characterization  of permutations $\sigma$ such that ${\Lambda}_{\sigma}$ are invariant under $\Phi$. We also consider the invariant principal order  ideals with respect to the Bruhat order  under Han's bijection $H$. We find  that ${\Lambda}_{\sigma}$ is invariant under the bijection $H$ if and only if it is invariant under the transformation $\Phi$.

scholarly journals THE EXPECTED JAGGEDNESS OF ORDER IDEALS

2017 ◽  
Vol 5 ◽  
Author(s):  
MELODY CHAN ◽  
SHAHRZAD HADDADAN ◽  
SAM HOPKINS ◽  
LUCA MOCI
Keyword(s):  
Probability Distribution ◽  
Young Diagram ◽  
Partially Ordered Sets ◽  
Order Ideal ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Combinatorial Theorem ◽  
Order Ideals ◽  
Geometric Formula

Thejaggednessof an order ideal$I$in a poset$P$is the number of maximal elements in$I$plus the number of minimal elements of$P$not in$I$. A probability distribution on the set of order ideals of$P$istoggle-symmetricif for every$p\in P$, the probability that$p$is maximal in$I$equals the probability that$p$is minimal not in$I$. In this paper, we prove a formula for the expected jaggedness of an order ideal of $P$under any toggle-symmetric probability distribution when$P$is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015,arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

The Effects of Drying Techniques on Clay-Rich Soil Texture

1974 ◽  
Vol 32 ◽  
pp. 466-467
Author(s):  
T. G. Naymik
Keyword(s):  
Critical Point ◽  
Liquid Nitrogen ◽  
Liquid Nitrogen Temperature ◽  
Drying Methods ◽  
Air Drying ◽  
Freeze Dried ◽  
Critical Point Drying ◽  
Set Up ◽  
The Relationship ◽  
Quartz Grains

Three techniques were incorporated for drying clay-rich specimens: air-drying, freeze-drying and critical point drying. In air-drying, the specimens were set out for several days to dry or were placed in an oven (80°F) for several hours. The freeze-dried specimens were frozen by immersion in liquid nitrogen or in isopentane at near liquid nitrogen temperature and then were immediately placed in the freeze-dry vacuum chamber. The critical point specimens were molded in agar immediately after sampling. When the agar had set up the dehydration series, water-alcohol-amyl acetate-CO2 was carried out. The objectives were to compare the fabric plasmas (clays and precipitates), fabricskeletons (quartz grains) and the relationship between them for each drying technique. The three drying methods are not only applicable to the study of treated soils, but can be incorporated into all SEM clay soil studies.

Evaluation of Freezing Methods by Low Temperature X-Ray Diffraction

1977 ◽  
Vol 35 ◽  
pp. 330-333
Author(s):  
T. Gulik-Krzywicki ◽  
M.J. Costello
Keyword(s):  
Biological Materials ◽  
Molecular Organization ◽  
X Ray Diffraction ◽  
Glass Capillary ◽  
X Ray ◽  
Rate Dependent ◽  
Before And After ◽  
Freeze Etching ◽  
Diffraction Patterns ◽  
Set Up

Freeze-etching electron microscopy is currently one of the best methods for studying molecular organization of biological materials. Its application, however, is still limited by our imprecise knowledge about the perturbations of the original organization which may occur during quenching and fracturing of the samples and during the replication of fractured surfaces. Although it is well known that the preservation of the molecular organization of biological materials is critically dependent on the rate of freezing of the samples, little information is presently available concerning the nature and the extent of freezing-rate dependent perturbations of the original organizations. In order to obtain this information, we have developed a method based on the comparison of x-ray diffraction patterns of samples before and after freezing, prior to fracturing and replication.Our experimental set-up is shown in Fig. 1. The sample to be quenched is placed on its holder which is then mounted on a small metal holder (O) fixed on a glass capillary (p), whose position is controlled by a micromanipulator.

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