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The alternating sign matrices and descending plane partitions : n+3 pairs of equivalent statistics
Virtualconference
Authors : Fischer, Ilse (Author of the conference)
CIRM (Publisher )
05A05 - Permutations, words, matrices
05A15 - Exact enumeration problems, generating functions
05A19 - Combinatorial identities, bijective combinatorics
82B20 - Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B23 - Exactly solvable models; Bethe ansatz
15B35 - Sign pattern matrices
Additional resources :
https://www.cirm-math.fr/RepOrga/2324/Slides/slides_fischer.pdf
CIRM (Publisher )
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Abstract :
There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is, despite many efforts, an open problem for about $40$ years now. So far, four pairs of statistics that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with $n+3$ pairs of statistics that have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the $(-1)$enumerations of these extended objects with respect to one pair of the $n+3$ pairs of statistics. One important tool of our proof is a multivariate generalization of the operator formula for the number of monotone triangles with prescribed bottom row that generalizes Schur functions. Joint work with Florian Aigner.
Keywords : Combinatorics; Mathematical Physics
alternating sign matrices; descending plane partitions; constant term identities; symmetric functions; bijective proofs
MSC Codes : Keywords : Combinatorics; Mathematical Physics
alternating sign matrices; descending plane partitions; constant term identities; symmetric functions; bijective proofs
05A05 - Permutations, words, matrices
05A15 - Exact enumeration problems, generating functions
05A19 - Combinatorial identities, bijective combinatorics
82B20 - Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B23 - Exactly solvable models; Bethe ansatz
15B35 - Sign pattern matrices
Additional resources :
https://www.cirm-math.fr/RepOrga/2324/Slides/slides_fischer.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 02/08/2021 Conference Date : 25/06/2021 Subseries : Research talks arXiv category : Combinatorics ; Mathematical Physics Mathematical Area(s) : Combinatorics ; Mathematical Physics Format : MP4 (.mp4) - HD Video Time : 00:53:31 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-06-25_Fisher.mp4 
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Information on the Event
Event Title : Lattice Paths, Combinatorics and Interactions / Marches aléatoires, combinatoire et interactionsEvent Organizers : Banderier, Cyril ; Dousse, Jehanne ; Duchi, Enrica ; Krattenthaler, Christian ; Wallner, Michael Dates : 21/06/2021 - 25/06/2021 Event Year : 2021 Event URL : https://conferences.cirm-math.fr/2324.html
Citation Data
DOI : 10.24350/CIRM.V.19770503Cite this video as: Fischer, Ilse (2021). The alternating sign matrices and descending plane partitions : n+3 pairs of equivalent statistics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19770503 URI : http://dx.doi.org/10.24350/CIRM.V.19770503 |
See Also
- [Virtualconference] Using symbolic dynamical programming in lattice paths combinatorics / Author of the conference Zeilberger, Doron.
- [Multi angle] Lattice paths and heaps / Author of the conference Viennot, Xavier.
- [Multi angle] Mating of discrete trees and walks in the quarter plane / Author of the conference Biane, Philippe.
- [Virtualconference] Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude / Author of the conference Marcovici, Irène.
- [Virtualconference] Random triangulations and bijective paths to Liouville quantum gravity / Author of the conference Holden, Nina.
Bibliography
- ANDREWS, George E. Plane partitions (III): The weak Macdonald conjecture. Inventiones mathematicae, 1979, vol. 53, no 3, p. 193-225. - https://doi.org/10.1007/BF01389763
- MILLS, William H., ROBBINS, David P., et RUMSEY JR, Howard. Alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A, 1983, vol. 34, no 3, p. 340-359. - https://doi.org/10.1016/0097-3165(83)90068-7
- ROBBINS, David P. et RUMSEY JR, Howard. Determinants and alternating sign matrices. Advances in Mathematics, 1986, vol. 62, no 2, p. 169-184. - https://doi.org/10.1016/0001-8708(86)90099-X
- ZEILBERGER, Doron. Proof of the alternating sign matrix conjecture. arXiv preprint math/9407211, 1994. - https://arxiv.org/abs/math/9407211
- AIGNER, Florian et FISCHER, Isle. The relation between alternating sign matrices and descending plane partitions: n+3 pairs of equivalent statistics. arXiv preprint math/2106.11568, 2021. - https://arxiv.org/abs/2106.11568
