Geodesic
The Hopf algebra of uniform block permutations.
Journal of Algebraic Combinatorics, Tome 28 (2008) no. 1, pp. 115-138.

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Summary: We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Rosas, and Sagan. These two embeddings correspond to the factorization of a uniform block permutation as a product of an invertible element and an idempotent one.
Keywords: keywords Hopf algebra, factorizable inverse monoid, uniform block permutation, set partition, symmetric functions, Schur-Weyl duality 
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Aguiar, Marcelo; Orellana, Rosa C. The Hopf algebra of uniform block permutations.. Journal of Algebraic Combinatorics, Tome 28 (2008) no. 1, pp. 115-138. http://geodesic.mathdoc.fr/item/JAC_2008__28_1_a5/