General Information
| Student: | Volodymyr Kuznietsov |
|---|---|
| Office: | CoRE 448 |
| School: | Faculty of Mathematics and Physics, Charles University |
| Contact: | kuzvladim7@gmail.com |
| Project: | k-Stanley inequality |
| Collaborators: | S. Kotsiubynska |
| Mentor: | Swee Hong Chan |
I am a part of a group of students from Charles University that includes Ben Bencik, Adam Dzavoronok, Guillermo Gamboa, Jelena Glisic, Robert Jaworski, Tymur Kotkov, Todor Antić, Julia Krizanova, Volodymyr Kuznietsov, Tymofii Reizin, Jakub Sosovicka, Filip Uradnik, and Patrik Zavoral.
Project Description
- Fix an element z ∈ P and define Nk as the number of linear extensions L, which satisfy L(z)=k for every k ∈ [n]. In 1981 Stanley proved that for every poset P the sequence Nk2 satisfy Nk2 ≥N k-1Nk+1 for 1<k<n, k ∈ N. The k-Stanley inequality is kNk2 ≥ (k-1)Nk-1Nk+1+NkNk+1, it is a partial case of Cross-product conjecture can extend Stanley inequality, if it is true.
Research Log
Week 1 (05/28-06/02)
- We arrived to Rutgers. In two days we had a meeting with our supervisor, where we discussed a problem and first steps how to work on it. We adapted to the place. On Sunday we went to New York with a part of our group and had a great time together.
Week 2 (06/03-06/09)
- We have proved cases for k=1 and k=2, which were quite trivial. Also, we proved a conjecture when N1 is not 0, using unimodality, so now we consider linear extensions for which N1 = 0. Also, went to New York again to meet with our friend. We visited the Metropolitan Museum, which was the second time for me there. However, we saw a lot of new things and great paintings.
Week 3 (06/10-06/16)
- We tried to think in other direction, so we assumed that k-Stanley is true and we got a pretty interesting implication. Assuming k-Stanley is true and that N1 = ... = Na = 0, we get that for any k greater equal than a, kNk >= (k-a)Nk+1. We will try to find counterexample for this, which immedietly disproves k-Stanley, or show that it is true.
Week 4 (06/17-06/23)
- We wrote a program to count the number of valid linear extensions. Firstly, it worked slowly for even width-2 posets. Then the supervisor Swee Hong gave a hint to implement this program using a dynamic programming for width-2 and width-3 posets. We used updated code to find any counterexamples to the conjecture made last week, but everything was in vain.
Week 5 (06/24-06/30)
Acknowledgements
- This work was carried out while the author Volodymyr Kuznietsov was a participant in the 2024 DIMACS REU program,
supported by CoSP, a project funded by European Union’s Horizon 2020 research and innovation programme, grant
agreement No. 823748.