Anders Claesson: Caylerian Polynomials

Consider a ballot (ordered set partition) such as {2,3,5}{6}{1,7}{4}. It may be represented by a function, say v, such that i belongs to the v(i)-th block. In our case, v(1)=3, v(2)=1, v(3)=1, etc. This type of function is sometimes referred to as a Cayley permutation. That is, a Cayley permutation is a function from [n] to [n] whose image is an initial segment [k] of [n].

We have begun to explore descent polynomials over Cayley permutations. By establishing a connection to so-called Burge words and Burge matrices we derive some results that parallel classical theorems about Eulerian polynomials. Furthermore, we reformulate the gamma-nonnegativity of the two-sided Eulerian polynomials in terms of these Burge structures.

This is joint work with Giulio Cerbai (University of Iceland).