  We investigate the $K$-theoretic Gysin map for type $A$ partial flag bundles
from the viewpoint of integrability. We introduce several types of partition
functions for one version of $q=0$ degeneration of $U_q(\widehat{sl_n})$ vertex
models on rectangular grids which differ by boundary conditions and sizes, and
can be viewed as Grothendieck classes of the Grothendieck group of a
nonsingular variety and partial flag bundles. By deriving multiple commutation
relations for the $q=0$ $U_q(\widehat{sl_n})$ Yang-Baxter algebra and combining
with the description of the $K$-theoretic Gysin map for partial flag bundles
using symmetrizing operators, we show that the $K$-theoretic Gysin map of the
first type of partition functions on a rectangular grid is given by the second
type whose boundary conditions on one side are reversed from the first type.
This generalizes the author's previous result from Grassmann bundles to partial
flag bundles. We also discuss the inhomogenous version of the partition
functions and applications to the $K$-theoretic Gysin map.
