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Hopf algebra of permutations
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<p>In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a> with a basis of all elements of all the finite symmetric groups <i>S</i><i>n</i>, and is a non-commutative analogue of the <a href="/facts/Hopf_algebra_of_symmetric_functions/SmENv9o4">Hopf algebra of symmetric functions</a>. It is both <a href="/facts/Free_algebra/xMrT0WbT">free</a> as an <a href="/facts/Algebra_(ring_theory)/8T8ulWJb">algebra</a> and graded-<a href="/facts/Cofree_coalgebra/CV5tkp4q">cofree</a> as a graded <a href="/facts/Coalgebra/pvx52xS6">coalgebra</a>, so is in some sense as far as possible from being either commutative or cocommutative. It was introduced by Malvenuto & Reutenauer (1995) and studied by Poirier & Reutenauer (1995). </p>