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.

scholarly journals Enumeration of Alternating Sign Matrices of Even Size (Quasi-)Invariant under a Quarter-Turn Rotation

10.37236/323 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jean-Christophe Aval ◽  
Philippe Duchon
Keyword(s):  
Alternating Sign Matrices ◽  
Enumeration Formula ◽  
Half Turn

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 unrestricted ASM's and the number of half-turn symmetric ASM's.

scholarly journals A bijection for nonorientable general maps

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jérémie Bettinelli
Keyword(s):  
Simple Form ◽  
Asymptotic Enumeration ◽  
International Audience ◽  
Enumeration Formula

International audience We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.

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 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 On trees, tanglegrams, and tangled chains

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
Frderick Matsen IV
Keyword(s):  
Asymptotic Formula ◽  
Computer Science ◽  
Explicit Formula ◽  
Binary Trees ◽  
Biological Research ◽  
Open Problems ◽  
International Audience ◽  
Enumeration Formula ◽  
New Formula ◽  
Simple Asymptotic Formula

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

scholarly journals The Selberg integral and Young books

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jang Soo Kim ◽  
Suho Oh
Keyword(s):  
Young Tableaux ◽  
Combinatorial Interpretation ◽  
Selberg Integral ◽  
Combinatorial Objects ◽  
Special Cases ◽  
International Audience ◽  
Enumeration Formula ◽  
Standard Young Tableaux

International audience The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained. L’intégrale de Selberg est une partie intégrante importante abord évaluée par Selberg en 1944. Stanley a trouvé une interprétation combinatoire de la Selberg aide en permutations. Dans ce papier, de nouveaux objets combinatoires “livres de Young” sont introduits et présentés à avoir un lien avec l’intégrale de Selberg. Cette connexion donne une formule d'énumération pour les livres de Young. Il est démontré que des cas spéciaux de livres de Young deviennent tableaux standards de Young de formes diverses: escaliers décalés, places, certaines formes gauches et certaines formes tronquées. En conséquence, l’énumération des formules pour tableaux standards de Young de ces formes sont obtenues.

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 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.

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