General Information

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Project Description

• Fix an element z ∈ P and define N_k 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 N_k² satisfy N_k² ≥N _k-1N_k+1 for 1<k<n, k ∈ N. The k-Stanley inequality is kN_k² ≥ (k-1)N_k-1N_k+1+N_kN_k+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 = ... = N_a = 0, we get that for any k greater equal than a, kN_k >= (k-a)N_k+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)

We did not have a lot of progress this week. I was checking a lot of examples with my code and they all satisfied the conjecture. I began to believe that conjecture is true. Also I had a wonderful trip to New York. This time I have visited a lot of new places and also I have been to the Ukrainian restaurant in Manhattan which I liked very much.<\ul>

Week 6 (07/01-07/07)

My supervisor returned back after 3 weeks spent in China and Canada, where he gave talks at the universities. I have shared my results with him and he paid a lot of attention to the inequalities derived from k-Stanley if it's true. Basically, it gives us an upper and lower bound for the ratio between consecutive N_k. Started to look for counterexamples of this, but could not find one. I spent weekends in Philadelphia and Washington D.C.. On Saturday I went to visit my friend in Philadelphia and in the evening I was visiting my relatives who live nearby. The next day we went to Washington D.C. and spent couple hours there. The trip was long, so we made it when it was middle of the day and the sun was very very hot. We were waslking around White House, Washington Monument, Capitol and Lincoln memorial. All of them seem to be close to each other, but in reality thay don't and it was hard to survive under this sun. Then I returned back to Rutgers and was preparing for the next week.

Week 7 (07/08 - 07/14)

We tried to prove the conjecture inductively, using some sums of posets. These sums can not construct all posets, but a big a range of them. We considered parallel and series sum. For series sum the conjecture is trivial, so we made a little theorem. For the parralel sum it becomes harder and we could prove only if you add one element. We tried to prove if we add a chain or a chain of 2 elements, but it's complicated. This week I have been to New York and Philadelphia meeting with my friends. I visited American Museum of Natural History, which was amazing. I was surprised by looking at so realistic copies of animals and historical events. Except this, I had a wonderful time with my friends.

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.