On Combinatorial Bijection between Alternating Sign Matrices and Plane Partitions

Student: Babina Nataliia

Supervisor: Takashi Takebe

Educational Programme: Mathematics (Bachelor)

Final Grade: 9

Year of Graduation: 2020

Descending plane partitions and alternating sign matrices are combinatorial objects that have been proven to have closely related generating functions. However, a direct combinatorial bijection between these sets has not been found yet. We prove the formula binding the inversion number of an alternating sign matrix, the number of $-1$'s and the number of diagonal equalities in the corresponding monotone triangle. Also we present the algorithm and the program generating corresponding subsets of descending plane partitions and alternating sign matrices which can be used to study examples and verify some assumptions about the bijection. Further we define and research an infinite tree generated by nested monotone triangles and descending plane partitions. Finally we research some specific examples and their properties.

Full text (added June 1, 2020)

Student Theses at HSE must be completed in accordance with the University Rules and regulations specified by each educational programme.

Summaries of all theses must be published and made freely available on the HSE website.

The full text of a thesis can be published in open access on the HSE website only if the authoring student (copyright holder) agrees, or, if the thesis was written by a team of students, if all the co-authors (copyright holders) agree. After a thesis is published on the HSE website, it obtains the status of an online publication.

Student theses are objects of copyright and their use is subject to limitations in accordance with the Russian Federation’s law on intellectual property.

In the event that a thesis is quoted or otherwise used, reference to the author’s name and the source of quotation is required.

Search all student theses