scholarly journals The Many Faces of Alternating-Sign Matrices

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
James Propp
Keyword(s):  
Height Function ◽  
Original Form ◽  
Alternating Sign Matrices ◽  
International Audience ◽  
Order Ideals ◽  
The Many

International audience I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.

scholarly journals Promotion and Rowmotion

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jessica Striker ◽  
Nathan Williams
Keyword(s):  
Recent Work ◽  
Linear Extensions ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
International Audience ◽  
Order Ideals

International audience We present an equivariant bijection between two actions—promotion and rowmotion—on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions. Nous prèsentons une bijection èquivariante entre deux actions—promotion et rowmotion—sur les idèaux d'ordre dans certaines posets. Cette bijection gènèralise simultanèment un rèsultat de R. Stanley concernant la promotion sur les extensions linèaire de deux cha\^ınes disjointes et certains cas des travaux rècents de D. Armstrong, C. Stump, et H. Thomas sur les partitions noncroisèes et nonembo\^ıtèes. Nous appliquons cette bijection à plusieurs classes de posets pour obtenir des bijections èquivariantes a des diffèrents objets connus sous la rotation. Nous gènèralisons la même idèe pour donnè une bijection èquivariante entre les matrices à signes alternants sous rowmotion et sous la gyration de B. Wieland. Finalement, nous dèfinissons deux actions avec des ordres similaires sur les matrices à signes alternants et les partitions plane totalement symètriques et autocomplèmentaires.

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 Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Christophe Aval ◽  
Philippe Duchon
Keyword(s):  
Alternating Sign Matrices ◽  
International Audience ◽  
Enumeration Formula ◽  
Half Turn

International audience The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestrited ASm's and the number of half-turn symmetric ASM's. L'objet de ce travail est d'énumérer les matrices à signes alternants (ASM) quasi-invariantes par rotation d'un quart-de-tour. La formule d'énumération, conjecturée par Duchon, fait apparaître trois facteurs, comprenant le nombre d'ASM quelconques et le nombre d'ASM invariantes par demi-tour.

Recent Advances in Quantitative X-Ray Spectrometric Analysis by Solution Techniques

1967 ◽  
Vol 11 ◽  
pp. 1-22
Author(s):  
Eugene P. Bertin
Keyword(s):  
Matrix Effects ◽  
Internal Standard ◽  
Standard Addition ◽  
Fusion Product ◽  
Spectrometric Analysis ◽  
Original Form ◽  
Liquid Form ◽  
X Ray ◽  
The Many ◽  
Solution Techniques

AbstractUsually X-ray spectrometric analyse? of samples in their original forms—solid, powder, small fabricated parts, liquid, etc.—are more rapid and convenient than analyses by any other method. Thus, the analyst is well advised to strive to analyze samples in their original form whenever practical. However, it is often necessary to reduce nonliquid samples to some other form, for example, when standards are unavailable in the original form, when the same substance is received for analysis in a variety of forms, when homogeneity or matrix effects are severe, or when an internal standard must be added. In such cases, samples and standards are often reduced to a powder, a fusion product, or a solution. If the decision is made to put the sample into solution, the many well-known advantages of solution techniques are realised, including: (1) homogeneity; (2) easy preparation of standards and blanks; (3) easy concentration, dilution, separation, and other treatment; (4) reduced matrix effects and wide choice of ways to deal with matrix effects; (5) wide choice of ways to present the specimen to the spectrometer; and (6) applicability of internal-standard, standard-addition or -dilution, indirect, absorption, and scatter methods. This paper reviews work reported since 1960 in which the specimen is presented to the spectrometer in liquid form. The review is not particularly critical and stresses experimental techniques and treatment of data rather than specific materials and results. The work is discussed in the following categories: (1) sensitivity ; (2) liquid-specimen cells; (3) interaction of primary beam and liquid specimens; (4) matrix effects; (5) indirect (association) analysis; (6) X-ray scatter methods; and (7) X-ray absorption-edge spectrometry.

scholarly journals Boolean complexes and boolean numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Coxeter Group ◽  
Bruhat Order ◽  
Graph Invariant ◽  
Coxeter System ◽  
Combinatorial Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Order Ideals ◽  
Simplicial Poset ◽  
The Ideal

International audience The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.

The old temple terrace at the Argive Heraeum and the early cult of Hera in the Argolid

1982 ◽  
Vol 102 ◽  
pp. 186-201 ◽  
Author(s):  
James C. Wright
Keyword(s):  
Crucial Role ◽  
Recent Article ◽  
Original Form ◽  
Architectural Form ◽  
History Of ◽  
Construction Inspection ◽  
The Many ◽  
Architectural Organization

A recent article by Dr H. Plommer (‘Shadowy Megara’, JHS xcvii [1977] 75–88) has once again brought to our attention one of the many unresolved architectural problems at the Argive Heraeum—the date of the megalithic terrace on which the archaic temple was built. This terrace has been variously assigned to the Mycenaean, Geometric and Archaic periods and its role in the foundation of the cult has never been ascertained. In view of this continuing lack of consensus among modern scholars and the murkiness of the history of the origins of the Hera sanctuary, a restatement and re-examination of the evidence are in order. In this article I will first consider the date of the terrace and then attempt to place it in the perspective of early cult activity in the Argolid. This will require a survey of the proposed dates for the terrace and a close look at the remains of the Archaic Hera temple and its stratigraphic and architectural relation to the terrace. An inquiry into the form of the terrace will lead to an explanation of its unique architectural form and to a hypothesis for the reason for its construction. Inspection of the remains and a reconstruction of the original form of other early cult centers, notably Mycenae and Tiryns, will provide a context for understanding the origin and architectural form of the early Heraeum. In conclusion I will suggest that the presence of Mycenaean monuments in the Argolid, more than elsewhere, played a crucial role in the formation and architectural organization of the principal cults.

scholarly journals Diagonally and antidiagonally symmetric alternating sign matrices of odd order

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Roger Behrend ◽  
Ilse Fischer ◽  
Matjaz Konvalinka
Keyword(s):  
Partition Function ◽  
General Expression ◽  
Schur Function ◽  
Vertex Model ◽  
Grid Graph ◽  
Alternating Sign Matrices ◽  
Odd Order ◽  
International Audience ◽  
Reflection Equations

International audience We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

scholarly journals Piecewise-linear and birational toggling

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
David Einstein ◽  
James Propp
Keyword(s):  
Symmetry Property ◽  
Piecewise Linear ◽  
Young Tableaux ◽  
Fonction Transfert ◽  
International Audience ◽  
Reciprocal Symmetry ◽  
Transfer Map ◽  
Order Ideals

International audience We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset $P$ as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When $P = [a] \times [b]$, a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. We prove some homomesy results, showing that for certain functions $f$, the average of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen. Nous définissons et étudions certains analogues linéaires-par-morceaux et birationnels d’involutions toggles, rowmotion et promotion sur les idéaux d’un poset $P$, comme étudié par Striker et Williams. La rowmotion linéaire-par-morceaux est liée à la fonction transfert de Stanley pour les polytopes d’ordre; la promotion linéaire-par-morceaux se rapporte à la promotion de Schützenberger pour les tableaux semi-standards de Young. Lorsque $P = [a] \times [b]$, une propriété de symétrie réciproque récemment prouvée par Grinberg et Roby implique que la rowmotion birationnelle (et par conséquent la rowmotion linéaire-par-morceaux) est de l’ordre $a+b$. Nous démontrons quelques résultats d’homomésie, montrant que pour certaines fonctions $f$, la moyenne de $f$ sur chaque orbite de rowmotion/promotion est indépendante de l’orbite choisie.

scholarly journals Generalized monotone triangles

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lukas Riegler
Keyword(s):  
Recent Work ◽  
Computational Experiments ◽  
Combinatorial Interpretation ◽  
Alternating Sign Matrices ◽  
Round Numbers ◽  
International Audience

International audience In a recent work, the combinatorial interpretation of the polynomial $\alpha (n; k_1,k_2,\ldots,k_n)$ counting the number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ was extended to weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n$. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles – a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of $\alpha (n; k_1,k_2,\ldots,k_n)$ at arbitrary $(k_1,k_2,\ldots,k_n) ∈ \mathbb{Z}^n$ is a signed enumeration of Generalized Monotone Triangles with bottom row $(k_1,k_2,\ldots,k_n)$. Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving bijective proofs.

scholarly journals Extended galleries above the porch in two mosques: Qualitative analysis of mosques with wooden minaret in Bosnia and Herzegovina

2021 ◽  
Vol 3 (2) ◽  
pp. 78-88
Author(s):  
Edin Jahic
Keyword(s):  
Qualitative Analysis ◽  
Bosnia And Herzegovina ◽  
Rural Areas ◽  
Original Form ◽  
Common Features ◽  
Spatial Concept ◽  
The Common ◽  
The Many ◽  
Typical Solution

Among the many mosques from the Ottoman period in Bosnia and Herzegovina, the most numerous are modest and predominantly wooden mosques covered by a hip roof with an integrated wooden minaret. Although they originate in the long tradition of Turkish single-space mosques, their appearance and construction represent the expression of Bosnian autochthonous architecture. They were mostly built for the needs of the neighborhood (mahala) in smaller and larger towns, but also in rural areas. Due to the perishable materials and various other reasons, they had been renovated several times so changes in appearance were in some cases quite certain. These structures have been insufficiently researched and very few valuable publications are available so far. Qualitative analysis of significant examples, in addition to the common features by which these mosques differ from large monumental mosques, differences in the spatial concept, as well as the construction of individual elements, were observed. Concerning the shape of the entrance, these mosques have four characteristic solutions: a mosque with a porch, with a porch and a gallery, without a porch, and with a closed vestibule. The analysis also showed that the two mahala mosques in Tuzla had a specific gallery form that deviated from the typical solution. These galleries are extended over the porch on three sides by the application of ingenious carpentry solutions and covered with elongated eaves. In addition, this study showed that thanks to available sources, it was possible to re-establish the original form of the two mosques, which had since been altered.

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