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Colin Defant: Fertilitopes

Time: Wed 2021-04-14 15.15 - 16.15

Location: Zoom meeting ID: 654 5562 3260

Participating: Colin Defant (Princeton University)

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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:
Zoom meeting ID: 654 5562 3260