  Introduced by Solomon in his 1976 paper, the descent algebra of a finite
Coxeter group received significant attention over the past decades. As proved
by Gessel, in the case of the symmetric group its structure constants give the
comultiplication table for the fundamental basis of quasisymmetric functions.
We show that this property actually implies several well known relations linked
to the Robinson-Schensted-Knuth correspondence and some of its generalisations.
We further use the theory of type B quasisymmetric functions introduced by Chow
to provide analogue results when the Coxeter group is the hyperoctahedral
group.
