A New Descent Algebra for Standard Parabolic Subgroups of $W(A_n)$

A new descent algebra $\sum_{W}(A_{n})$ of Weyl groups of type $A_n$, constructed by present authors in [1], is generated by equivalence classes $[x_J]$ arising from the equivalence relation defined on the set of all $x_J$. In this paper, we introduce the structure of this new descent algebra for standard parabolic subgroups $W_J$ of Weyl groups of type $A_n$.