Petrie symmetric functions

For any positive integer k and nonnegative m, we consider the symmetric function Gk,m defined as sum of all monomials degree m that involve only exponents smaller than k. We call a Petrie in honor Flinders Petrie, coefficients its expansion Schur basis are determinants matrices (and thus belong to 0,1,-1 by classical result Gordon Wilkinson). More generally, prove Pieri-like rule for expanding product form Gk,m·s μ whenever is partition; this 0,1,-1. also show Gk,1,Gk,2,Gk,3,... an algebraically independent generating set functions when 1-k invertible base ring, conjecture Liu Polo about Gk,2k-1 basis.