  Our main result here is that the specialization at $t=1/q$ of the $Q_{km,kn}$
operators studied in [4] may be given a very simple plethystic form. This
discovery yields elementary and direct derivations of several identities
relating these operators at $t=1/q$ to the Rational Compositional Shuffle
conjecture of [3]. In particular we show that if $m,n $ and $k$ are positive
integers and $(m,n)$ is a coprime pair then $$ q^{(km-1)(kn-1)+k-1\over 2}
Q_{km,kn}(-1)^{kn}\Big|_{t=1/q} \,=\, \textstyle{[k]_q\over [km]_q} e_{km}\big[
X[km]_q\big] $$ where as customarily, for any integer $s \geq 0$ and
indeterminate $u$ we set $[s]_u=1+u+\cdots +u^{s-1}$. We also show that the
symmetric polynomial on the right hand side is always Schur positive. Moreover,
using the Rational Compositional Shuffle conjecture, we derive a precise
formula expressing this polynomial in terms of Parking functions in the
$km\times kn$ lattice rectangle.
