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For any positive integer $k$ and nonnegative integer $m$, we consider thesymmetric function $G\left( k,m\right)$ defined as the sum of all monomials ofdegree $m$ that involve only exponents smaller than $k$. We call $G\left(k,m\right)$ a "Petrie symmetric function" in honor of Flinders Petrie, as thecoefficients in its expansion in the Schur basis are determinants of Petriematrices (and thus belong to $\left\{ 0,1,-1\right\} $ by a classical result ofGordon and Wilkinson). More generally, we prove a Pieri-like rule for expandinga product of the form $G\left( k,m\right) \cdot s_{\mu}$ in the Schur basiswhenever $\mu$ is a partition; all coefficients in this expansion belong to$\left\{ 0,1,-1\right\} $. We also show that $G\left( k,1\right) ,G\left(k,2\right) ,G\left( k,3\right) ,\ldots$ form an algebraically independentgenerating set for the symmetric functions when $1-k$ is invertible in the basering, and we prove a conjecture of Liu and Polo about the expansion of $G\left(k,2k-1\right)$ in the Schur basis.

Petrie symmetric functions