I know that many examples of Hopf algebras that come from combinatorics. But I'm interested in knowing how Hopf algebras are applied in solving combinatorial problem.
Are there examples of open problems in combinatorics which were finally solved using techniques from Hopf algebras?
Or where Hopf Algebra has been used to get alternative proofs of already known combinatorial results?
I can't answer your question but one person who is probably only using Hopf Algebras for applications to combinatorics is Persi Diaconis.
See a nice paper by Diaconis, Pang and Ram here: http://arxiv.org/pdf/1206.3620.pdf