Speaker< span style="font-variant: normal;">: Alejandro Morales (Montreal)
Title: The generalized Pitman-Stanley flow polytope
Abstract: In 1999, Pitman and Stanley introdu ced the polytope bearing their name along with a study of its faces, lattic e points, and volume. This polytope is well-studied due to its connections to parking functions, lattice path matroids, generalized permutahedra/polym atroids, and flow polytopes. Its lattice points correspond to plane partiti ons of skew shape with entries 0 and 1. Pitman and Stanley remarked that th eir polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries 0,1,...,m. Since then, this generaliz ation has been untouched. We study this polytope and show that it can also be realized as a flow polytope of a grid graph. In this talk I will discuss characterizations of its vertices and give formulas for the number of vert ices and faces as well as old and new formulas for the number of lattice po ints and volume in terms of rectangular Standard Young Tableaux. The new fo rmulas come from the volume polynomial formulas of flow polytopes in terms of vector partition functions of Baldoni and Vergne and lattice point formu las of Stanley of marked order polytopes.
This is joint work with Maura Hegarty , William Dugan, and Annie Raymond.
X-ALT-DESC;FMTTYPE=text/html:Speaker: Alejandro Morales ( Montreal)
Title: The generalized Pitman -Stanley flow polytope
Abstract: In 199
9, Pitman and Stanley introduced the polytope bearing their name along with
a study of its faces, lattice points, and volume. This polytope is well-st
udied due to its connections to parking functions, lattice path matroids, g
eneralized permutahedra/polymatroids, and flow polytopes. Its lattice point
s correspond to plane partitions of skew shape with entries 0 and 1. Pitman
and Stanley remarked that their polytope can be generalized so that lattic
e points correspond to plane partitions of skew shape with entries 0,1,...,
m. Since then, this generalization has been untouched. We study this polyto
pe and show that it can also be realized as a flow polytope of a grid graph
. In this talk I will discuss characterizations of its vertices and give fo
rmulas for the number of vertices and faces as well as old and new formulas
for the number of lattice points and volume in terms of rectangular Standa
rd Young Tableaux. The new formulas come from the volume polynomial formula
s of flow polytopes in terms of vector partition functions of Baldoni and V
ergne and lattice point formulas of Stanley of marked order polytopes.