Raziskovalni matematični seminar - Arhiv
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Datum in ura / Date and time: 17.6.21
(10:00 -- 11:00)
Predavalnica / Location: FAMNIT-VP1 & ZOOM
Predavatelj / Lecturer: Besfort Shala
Naslov / Title: The Probabilistic Zeta Function of a Finite Lattice
Vsebina / Abstract:
Let G be a finite group and let P(G, s) be the probability that an s-tuple of elements in G generate G. Due to Hall, there is a finite Dirichlet series expression for P(G, s), obtained by using the principle of Möbius inversion on partially ordered sets. Brown defined an analogous function P(L, s) for finite lattices, ultimately showing that P(C(G), s+1) = P(G, s), where C(G) is the coset lattice of the group G, that is, P(G, s) only depends on the coset lattice of G.
In this talk, we present our work as part of the "Final Project Paper" course, mentored by dr. Russ Woodroofe. We study Brown's definition of P(L, s), as well as propose a natural alternative that may be better-suited for non-atomic lattices. In this case, we obtain a general Dirichlet series expression for P(L, s), which need not be an ordinary Dirichlet series. We investigate general properties of P(L, s) and compute it on several examples of finite lattices, establishing connections with well-known identities. Furthermore, we investigate when P(L, s) is an ordinary Dirichlet series. Since P(C(G), s) is always such, we call such lattices coset-like. In this regard, we focus on partition lattices and 2-divisible partition lattices and show that they generally fail to be coset-like.
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