Speaker: Fabrizio Zanello, Michigan Tech
Title: Partitions with distinct parts and unimodality
Abstract: In this talk, we discuss the (non)unimodality of the rank-generating function Fλ(q) of the poset of partitions with distinct parts whose Ferrers diagrams are contained inside the Ferrers diagram of a given partition λ. This work, in collaboration with Richard Stanley, has in part been motivated by an attempt to place into a more general context the unimodality of Fλ(q) =(1+q)(1+q2)...(1+qn) , namely the rank-generating function associated to the "staircase" partition λ = (n, n-1, . . . , 1), for which determining a combinatorial proof remains an outstanding open problem to this day. Surprisingly, we will see that our type of results present some remarkable similarities to those shown in a 1990 paper by Dennis Stanton, who extended, to an arbitrary partition λ, the study of the unimodality of the q-binomial coefficient -i.e., the rank-generating function of the poset of arbitrary partitions whose Ferrers diagrams are contained inside a given rectangular Ferrers diagram. If time allows, we will also discuss a few recent developments on this topic, including a (prize-winning) paper by Levent Alpoge that has solved our conjecture on the unimodality of Fλ(q) when λ is the "truncated staircase" partition (n, n-1, . . . , n-(b-1)), for n much greater than b. We will conclude by mentioning several other conjectures or open problems.