** 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 F_{λ}(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 F_{λ}(q)
=(1+q)(1+q^{2})...(1+q^{n}) , 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 F_{λ}(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.