scholarly journals A Bijective Proof of the ASM Theorem, Part I: The Operator Formula

10.37236/9082 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Ilse Fischer ◽  
Matjaž Konvalinka
Keyword(s):  
Main Tool ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Bijective Proof

Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but no bijective proof for any of these equivalences has been found so far. In this paper we provide the first bijective proof of the operator formula for monotone triangles, which has been the main tool for several non-combinatorial proofs of such equivalences. In this proof, signed sets and sijections (signed bijections) play a fundamental role.

scholarly journals Chung-Feller Property of Schröder Objects

10.37236/5659 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Youngja Park ◽  
Sangwook Kim
Keyword(s):  
Connected Components ◽  
Combinatorial Interpretation ◽  
Noncrossing Partitions ◽  
Alternating Sign Matrices ◽  
Feller Property ◽  
Bijective Proof ◽  
Schröder Numbers ◽  
Schröder Paths

Large Schröder paths, sparse noncrossing partitions, partial horizontal strips, and $132$-avoiding alternating sign matrices are objects enumerated by Schröder numbers. In this paper we give formula for the number of Schröder objects with given type and number of connected components. The proofs are bijective using Chung-Feller style. A bijective proof for the number of Schröder objects with given type is provided. We also give a combinatorial interpretation for the number of small Schröder paths.

scholarly journals The quantum symmetricXXZchain at Δ = -½, alternating-sign matrices and plane partitions

2001 ◽  
Vol 34 (19) ◽  
pp. L265-L270 ◽  
Author(s):  
M T Batchelor ◽  
J de Gier ◽  
B Nienhuis
Keyword(s):  
Alternating Sign Matrices ◽  
Plane Partitions

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 Alternating sign matrices and descending plane partitions

1983 ◽  
Vol 34 (3) ◽  
pp. 340-359 ◽  
Author(s):  
W.H Mills ◽  
David P Robbins ◽  
Howard Rumsey
Keyword(s):  
Alternating Sign Matrices ◽  
Plane Partitions

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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