General Information
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 Nk 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. k-Stanley can extend Stanley inequality, if it is true.
Research Log
Week 1 (05/28-05/31)
We arrived at Rutgers University on May 28. The following day, we had an introduction day, and on May 30,
I met with my supervisor, Dr. Chan, for the first time. We discussed three problems, and I began my work by
familiarizing myself with them. On Saturday, we took a trip to New York City.
Week 2 (06/03-06/07)
First half of the second week was spent to read a paper about the Stanley inequality.
On Monday was a meeting with a supervisor where he explained how to make slides better, then on Tuesday was a presentation
of students' projects that they are going to work with. The second half of the week was spent on proving some trivial cases,
such as a case where N0 is not 0. The case N0 is 0, but all Ni where 1<i<k-1 are non-zero, gave a new
inequality that allows to disprove Stanley's inequality. I also tryed to apply lattice pathes to k-Stanley. Professor gave an exercise to
show how to obtain k-Stanley inequality from Cross-product conjecture.
The first half of the second week was dedicated to reading a paper on the Stanley inequality. On Monday, there was a meeting with my supervisor
who offered guidance on improving my slides for future presentation. The following day, Tuesday, featured presentations of student projects they planned
to work on. The latter half of the week involved proving some trivial cases, such as when N0 is not zero. The scenario where N0 is zero,
but all Ni for 1<i<k-1 are non-zero, led to a new inequality that could potentially disprove Stanley's inequality. I also attempted
to apply lattice paths to the k-Stanley inequality. The professor assigned an exercise to demonstrate how the k-Stanley inequality could be derived
from the Cross-product conjecture.
Week 3 (06/10-06/14)
We discussed the proof of the Stanley inequality and connection between the Cross-product conjecture and k-Stanley inequality with Dr. Chan.
The next step involved conducting simulations to identify a poset that might not satisfy the inequality. I spent Saturday and Sunday in New York
with my friend exploring some museums.
Week 4 (06/17-06/21)
This week we wrote a code and started checking the posets on width 2 and width 3.
Week 5 (06/24-06/28)
Simulations helped to understand the problem better and to obtain some new inequalities. We tried to check the inequality with simulations, but we could not find a counterexample. I checked posets on width 2 and I found some relations between different posets. This made me think that the conjecture is true for width 2 posets.
Week 6 (07/01-07/05)
The supervisor returned from his research trip back from China and Canada, we could discuss our results. We obtained an upper and lower bound for the ratio between consecutive Nk.
Week 7 (07/08-07/12)
We tried to prove the conjecture inductively, using some sums of posets, it could help to construct a big range of posets. Our supervisor recommended to consider parallel and series sum. For series sum the conjecture is trivial.
For the parralel sum we could prove only if you add one element since it is much harder. For the case when we add a 2-elements chain it was too complicated. On Monday we had a trip to Nokia bell labs, later that week I went on a boat trip around New York.
Week 8 (07/15-07/18)
This was the last week of the research, where we presented our results. We have not obtained any other useful results.
Acknowledgements
This work was carried out while the author Sofiia Kotsiubynska 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.
