# t-Stack Sorting on the Permutahedron

## Disciplines

Algebraic Geometry | Discrete Mathematics and Combinatorics | Geometry and Topology | Number Theory | Other Mathematics | Theory and Algorithms

## Abstract (300 words maximum)

In 1968, Knuth introduced the stack sorting algorithm which attempts to chronologically sort an inputted sequence, in our case a permutation. Using the stack sorting algorithm, we traverse the vertices and edges of the permutahedron. The n-permutahedron is the (n-1)-dimensional polytope generated by the convex hull of permutations of the first n natural numbers. We consider subpolytopes of the permutahedron arising from the convex hull of sequences generated by iterations of the stack sorting algorithm. For a particular family of subpolytopes, we determine their dimension and prove that they are simplices. We conjecture that this family of simplices has Ehrhart polynomials with coefficients in Pascal’s triangle.

## Academic department under which the project should be listed

CSM - Mathematics

## Primary Investigator (PI) Name

DR. JULIANNE VEGA

Andres R Vindas-Melendez, Mathematical Sciences Research Institute & Department of Mathematics at University of California Berkeley, avindas@msri.org

## Share

COinS

t-Stack Sorting on the Permutahedron

In 1968, Knuth introduced the stack sorting algorithm which attempts to chronologically sort an inputted sequence, in our case a permutation. Using the stack sorting algorithm, we traverse the vertices and edges of the permutahedron. The n-permutahedron is the (n-1)-dimensional polytope generated by the convex hull of permutations of the first n natural numbers. We consider subpolytopes of the permutahedron arising from the convex hull of sequences generated by iterations of the stack sorting algorithm. For a particular family of subpolytopes, we determine their dimension and prove that they are simplices. We conjecture that this family of simplices has Ehrhart polynomials with coefficients in Pascal’s triangle.