Till innehåll på sidan

# Colin Defant: Fertilitopes

Tid: On 2021-04-14 kl 15.15 - 16.15

Föreläsare: Colin Defant (Princeton University)

Abstract: The stack-sorting map is a combinatorially-defined operator s on the set of permutations of size n. The fertility of a permutation $$\pi$$ is the number of preimages of $$\pi$$ under the stack-sorting map. Associated to each permutation $$\pi$$ is a collection V($$\pi$$) of integer compositions, called the valid compositions of $$\pi$$. A crucial tool for understanding the stack-sorting map is the Fertility Formula, which expresses the fertility of $$\pi$$ as a sum over V($$\pi$$). Valid compositions also appear naturally in a formula in noncommutative probability theory that converts from free to classical cumulants. We define the fertilitope of $$\pi$$ to be the convex hull of V($$\pi$$). We will see that V($$\pi$$) is precisely the set of lattice points in the fertilitope of $$\pi$$. Moreover, fertilitopes have a surprisingly simple characterization as certain nestohedra obtained from binary plane trees. This implies that each set V($$\pi$$) is a discrete polymatroid, a fact that yields new information about the stack-sorting map. We will also discuss a conjecture about the real-rootedness of certain polynomials associated to the stack-sorting map. We will formulate an equivalent version of this conjecture in terms of nestohedra and suggest possible directions in which it could be extended.   Zoom link: https://kth-se.zoom.us/j/65455623260
Zoom meeting ID: 654 5562 3260
Innehållsansvarig:webmaster@math.kth.se
Tillhör: Institutionen för matematik