  In this paper it is proved that the ideal $I_w$ of the weak polynomial
identities of the superalgebra $M_{1,1}(E)$ is generated by the proper
polynomials $[x_1,x_2,x_3]$ and $[x_2,x_1][x_3,x_1][x_4,x_1]$. This is proved
for any infinite field $F$ of characteristic different from 2. Precisely, if
$B$ is the subalgebra of the proper polynomials of $F< X>$, we determine a
basis and the dimension of any multihomogeneous component of the quotient
algebra $B / B \cap I_w$. We compute also the Hilbert series of this algebra.
One of the main tools of the paper is a variant we found of the
Knuth-Robinson-Schensted correspondence defined for single semistandard
tableaux of double shape.
