Hopf algebra of permutations
In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the finite symmetric groups Sn. It is both free as an algebra and graded-cofree as a graded coalgebra, so in in some sense as for as possible from being either commutative or cocommutative. It was introduced by Malvenuto & Reutenauer (1994) and Poirier & Reutenauer (1995).