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

Tid: On 2023-03-22 kl 11.15 - 12.15

Plats: 3721 KTH

Medverkande: 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.