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 Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

10.37236/4971 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Partition Function ◽  
Lattice Paths ◽  
Schur Function ◽  
Vertex Model ◽  
Shifted Tableaux

A recent paper of Bump, McNamara and Nakasuji introduced a factorial version of Tokuyama's identity, expressing the partition function of  six vertex model as the product of a $t$-deformed Vandermonde and a Schur function. Here we provide an extension of their result by exploiting the language of primed shifted tableaux, with its proof based on the use of non-interesecting lattice paths.

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.

On some representations of the six vertex model partition function

2003 ◽  
Vol 315 (3-4) ◽  
pp. 231-236 ◽  
Author(s):  
F. Colomo ◽  
A.G. Pronko
Keyword(s):  
Partition Function ◽  
Vertex Model

Integrable equations for the partition function of the six-vertex model

2000 ◽  
Vol 100 (2) ◽  
pp. 2141-2146 ◽  
Author(s):  
A. G. Izergin ◽  
E. Karjalainen ◽  
N. A. Kitanin
Keyword(s):  
Partition Function ◽  
Vertex Model ◽  
Integrable Equations

scholarly journals Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end

2011 ◽  
Vol 844 (2) ◽  
pp. 289-307 ◽  
Author(s):  
Wen-Li Yang ◽  
Xi Chen ◽  
Jun Feng ◽  
Kun Hao ◽  
Bo-Yu Hou ◽  
...  
Keyword(s):  
Partition Function ◽  
Vertex Model ◽  
Determinant Formula

scholarly journals Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Masaki Watanabe
Keyword(s):  
Schur Function ◽  
Schubert Polynomials ◽  
International Audience

International audience We use the modules introduced by Kraśkiewicz and Pragacz (1987, 2004) to show some positivity propertiesof Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomialsis Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubertpolynomial is Schubert-positive. The present submission is an extended abstract on these results and the full versionof this work will be published elsewhere. Nous employons les modules introduits par Kraśkiewicz et Pragacz (1987, 2004) et démontrons certainespropriétés de positivité des polynômes de Schubert: nous donnons une nouvelle preuve pour le fait classique quele produit de deux polynômes de Schubert est Schubert-positif; nous démontrons aussi un nouveau résultat que lacomposition plethystique d’une fonction de Schur avec un polynôme de Schubert est Schubert-positif. Cet article estun sommaire de ces résultats, et une version pleine de ce travail sera publée ailleurs.

scholarly journals Congruences modulo powers of 5 for the rank parity function

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Dandan Chen ◽  
Rong Chen ◽  
Frank Garvan
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Theta Function ◽  
Mock Theta Function ◽  
Parity Function ◽  
International Audience

International audience It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

scholarly journals The polytope of Tesler matrices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales ◽  
Brendon Rhoades
Keyword(s):  
Partition Function ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Integer Points ◽  
International Audience ◽  
Kostant Partition Function ◽  
Tesler Matrices ◽  
Flow Polytope ◽  
Standard Young Tableaux

26 pages, 4 figures. v2 has typos fixed, updated references, and a final remarks section including remarks from previous sections International audience We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. We show that $Tes_n(a)$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of $Tes_n(a)$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of $Tes_n(1,1,...,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape. On présente le polytope de Tesler $Tes_n(a)$, dont les points réticuilaires sont les matrices de Tesler de taillen avec des sommes des équerres $a_1,a_2,...,a_n in Z_{\geq 0}$. On montre que $Tes_n(a)$ est un polytope de flux. Donc lenombre de matrices de Tesler est donné par la fonction de Kostant de type An évaluée à ($(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$On décrit les faces de ce polytope en termes de “tableaux de Tesler” et on caractérise quand le polytope est simple.On montre que l’h-vecteur de $Tes_n(a)$ , quand tous les $a_i>0$ , est donnée par le nombre de permutations avec unnombre donné d’inversions et on calcule le volume de T$Tes_n(1,1,...,1)$ comme un produit de nombres de Catalanconsécutives multiplié par le nombre de tableaux standard de Young en forme d’escalier

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