  We give some new formulas about factorizations of $K$-$k$-Schur functions
$g^{(k)}_{\lambda}$, analogous to the $k$-rectangle factorization formula
$s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}$ of $k$-Schur
functions, where $\lambda$ is any $k$-bounded partition and $R_t$ denotes the
partition $(t^{k+1-t})$ called \textit{$k$-rectangle}. Although a formula of
the same form does not hold for $K$-$k$-Schur functions, we can prove that
$g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t\cup\lambda}$, and in fact more generally
that $g^{(k)}_{P}$ divides $g^{(k)}_{P\cup\lambda}$ for any multiple
$k$-rectangles $P=R_{t_1}^{a_1}\cup\dots\cup R_{t_m}^{a_m}$ and any $k$-bounded
partition $\lambda$. We give the factorization formula of such $g^{(k)}_{P}$
and the explicit formulas of $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}$ in some
cases, including the case where $\lambda$ is a partition with a single part as
the easiest example.

Combinatorics