  A new combinatorial interpretation of the Howe dual pair
$(\hat{\frak{gl}}_{\infty|\infty},\frak{gl}_n)$ acting on an infinite
dimensional Fock space $\frak{F}^n$ of level $n$ is presented. The character of
a quasi-finite irreducible highest weight representation of
$\hat{\frak{gl}}_{\infty|\infty}$ occurring in $\frak{F}^n$ is realized in
terms of certain bitableaux of skew shapes. We study a general combinatorics of
these bitableaux, including Robinson-Schensted-Knuth correspondence and
Littlewood-Richardson rule, and then its dual relation with the rational
semistandard tableaux for $\frak{gl}_n$. This result also explains other Howe
dual pairs including $\frak{gl}_n$.
