Lattice Paths, Combinatorics and Interactions / Marches aléatoires, combinatoire et interactions

Collection Lattice Paths, Combinatorics and Interactions / Marches aléatoires, combinatoire et interactions

Organizer(s) Banderier, Cyril ; Dousse, Jehanne ; Duchi, Enrica ; Krattenthaler, Christian ; Wallner, Michael

Date(s) 21/06/2021 - 25/06/2021

linked URL https://conferences.cirm-math.fr/2324.html

00:00:00 / 00:00:00

3 6

The alternating sign matrices and descending plane partitions : n+3 pairs of equivalent statistics

By Ilse Fischer

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.

Information about the video

Date of recording 25/06/2021
Date of publication 02/08/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19770503
Cite this video Fischer, Ilse (25/06/2021). The alternating sign matrices and descending plane partitions : n+3 pairs of equivalent statistics. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19770503
URL https://dx.doi.org/10.24350/CIRM.V.19770503

Domain(s)

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

MSC codes

Document(s)

https://www.cirm-math.fr/RepOrga/2324/Slides/slides_fischer.pdf

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