R-S correspondence for the Hyper-octahedral group of type $B_n$ – A different approach

In this paper we develop a Robinson Schensted algorithm for the hyperoctahedral group of type $B_n$ on partitions of $(\frac{1}{2}r(r+1)+2n)$ whose $2-$core is $\delta_r$, $r\geq 0$ where $\delta_r$ is the partition with parts $(r,r-1,\dots,0)$. We derive some combinatorial properties associated with this correspondence.