# Leonardo Saud Maia Leite: A study about the chain polynomial of the lattice of flats of a matroid

**Time: **
Wed 2023-03-22 11.15 - 12.15

**Location: **
3721 KTH

**Participating: **
Leonardo Saud Maia Leite (KTH)

Abstract: The chain polynomial of a finite lattice \(\mathcal{L}\) is given by \(p_\mathcal{L} = \sum_{k ≥ 0} c_k (\mathcal{L}) x^k\), where \(c_k (\mathcal{L})\) is the number of chains of length \(k\) in \(\mathcal{L}\). There is a conjecture which states that, if \(\mathcal{L}\) is a geometric lattice, then its chain polynomial \(p_L\) is real-rooted. In particular, it is log-concave. Here, we will consider a finite matroid \(M\), define its lattice of flats \(L(M)\), and study the polynomial \(p_{L(M)}\). We verified that the conjecture is true for paving matroids and for some generalized paving matroids, a new class of matroids introduced during this study. This is an ongoing and joint work with Petter Brändén.