The probability of positivity in symmetric and quasisymmetric functions
probability, positivity, systemic functions, quasisymmetric functions
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.
Journal of Combinatorics