  We define a family of ideals $I_h$ in the polynomial ring
$\mathbb{Z}[x_1,...,x_n]$ that are parametrized by Hessenberg functions $h$
(equivalently Dyck paths or ample partitions). The ideals $I_h$ generalize
algebraically a family of ideals called the Tanisaki ideal, which is used in a
geometric construction of permutation representations called Springer theory.
To define $I_h$, we use polynomials in a proper subset of the variables
${x_1,...,x_n}$ that are symmetric under the corresponding permutation
subgroup. We call these polynomials {\em truncated symmetric functions} and
show combinatorial identities relating different kinds of truncated symmetric
polynomials. We then prove several key properties of $I_h$, including that if
$h>h'$ in the natural partial order on Dyck paths then $I_{h} \subset I_{h'}$,
and explicitly construct a Gr\"{o}bner basis for $I_h$. We use a second family
of ideals $J_h$ for which some of the claims are easier to see, and prove that
$I_h = J_h$. The ideals $J_h$ arise in work of Ding, Develin-Martin-Reiner, and
Gasharov-Reiner on a family of Schubert varieties called partition varieties.
Using earlier work of the first author, the current manuscript proves that the
ideals $I_h = J_h$ generalize the Tanisaki ideals both algebraically and
geometrically, from Springer varieties to a family of nilpotent Hessenberg
varieties.
