#### Title

The probability of positivity in symmetric and quasisymmetric functions

#### Department/School

Mathematics

#### Date

1-1-2018

#### Document Type

Article

#### Keywords

probability, positivity, systemic functions, quasisymmetric functions

#### Abstract

Given an element in a finite-dimensional real vector space, V, that is a nonnegative linear combination of basis vectors for some basis B, we compute the probability that it is furthermore a nonnegative linear combination of basis vectors for a second basis, A. We then apply this general result to combinatorially compute the probability that a symmetric function is Schur-positive (recovering the recent result of Bergeron--Patrias--Reiner), $e$-positive or $h$-positive. Similarly we compute the probability that a quasisymmetric function is quasisymmetric Schur-positive or fundamental-positive. In every case we conclude that the probability tends to zero as the degree of a function tends to infinity.

#### Volume

11

#### Issue

3

#### Published in

Journal of Combinatorics

## Citation/Other Information

Patrias, R., & van Willigenburg, S. (2018). The probability of positivity in symmetric and quasisymmetric functions. Journal of Combinatorics 11(3): 475-493. https://dx.doi.org/10.4310/JOC.2020.v11.n3.a3.