| 總結: | We give an elementary symmetric function expansion for the expressions
$M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$
and
$M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$
when
$t=1$
in terms of what we call
$\gamma $
-parking functions and lattice
$\gamma $
-parking functions. Here,
$\Delta _F$
and
$\Pi $
are certain eigenoperators of the modified Macdonald basis and
$M=(1-q)(1-t)$
. Our main results, in turn, give an elementary basis expansion at
$t=1$
for symmetric functions of the form
$M \Delta _{Fe_1} \Theta _{G} J$
whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis
$\{\Pi e_\lambda ^\ast \}_\lambda $
. Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.
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