  Weights of permutations were originally introduced by Dugan, Glennon,
Gunnells, and Steingr\'imsson (Journal of Combinatorial Theory, Series A
164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a
permutation $\sigma$ viewed as a sequence of integers, computing the weight of
$\sigma$ involves recursively counting descents of certain subpermutations of
$\sigma$. Using this weight function, one can define a $q$-analog $E_n(x,q)$ of
the Eulerian polynomials. We prove two main results regarding weights of
permutations and the polynomials $E_n(x,q)$. First, we show that the
coefficients of $E_n(x, q)$ stabilize as $n$ goes to infinity, which was
conjectured by Dugan, Glennon, Gunnells, and Steingr\'imsson (Journal of
Combinatorial Theory, Series A 164:24-49, 2019), and enables the definition of
the formal power series $W_d(t)$, which has interesting combinatorial
properties. Second, we derive a recurrence relation for $E_n(x, q)$, similar to
the known recurrence for the classical Eulerian polynomials $A_n(x)$. Finally,
we give a recursive formula for the numbers of certain integer partitions and,
from this, conjecture a recursive formula for the stabilized coefficients
mentioned above.
