Vertices of Generalized Pitman-Stanley Flow Polytopes

Presenter

Jose Miguel Cruz

Campus

UMass Amherst

Sponsor

Annie Raymond, Department of Mathematics and Statistics, UMass Amherst

Schedule

Session 2, 11:30 AM - 12:15 PM [Schedule by Time][Poster Grid for Time/Location]

Location

Poster Board A42, Campus Center Auditorium, Row 3 (A41-A60) [Poster Location Map]

Abstract

In 1999, Pitman and Stanley published a study related to an eponymous polytope. It is a well studied structure thanks to the many correspondences with combinatoric structures it possesses, such as parking functions, permutahedra, and plane partitions. To our interest, the lattice points of the polytope correspond to plane partitions of skew shape with entries in {0,1}. As a remark, Pitman and Stanley mentioned their polytope can be generalized such that the lattice points correspond to plane partitions with entries in {0,...,m}. Untouched for 24 years, Dugan, Hegarty, Morales and Raymond explored said generalization and concluded it can be realized as a flow polytope of a grid graph. They showed that the number of vertices of the polytopes are polynomials whose leading term sometimes count standard Young tableaux of skew shifted shape, and provided generating functions for the number of vertices of certain special cases. Using the SAGE programming language to generate these polytopes from their flow networks, manipulating their generating functions and exploiting the correspondence of vertices with skew shifted standard young tableaux, this paper aims to explore the enumeration of the vertices of these polytopes. Explicit formulas to count vertices are provided for some cases, and are conjectured for others.

Keywords

Eulerian Numbers, Catalan Numbers, Generating Functions, Plane Partitions, Young Tableaux

Research Area

Mathematics and Statistics

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