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THE HOPF ALGEBRA OF UNIFORM BLOCK PERMUTATIONS. EXTENDED ABSTRACT
 

Summary: THE HOPF ALGEBRA OF UNIFORM BLOCK PERMUTATIONS.
EXTENDED ABSTRACT
MARCELO AGUIAR AND ROSA C. ORELLANA
Abstract. 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.
RŽesumŽe. Nous prŽesentons l'alg`ebre de Hopf des permutations de blocs uniformes
est dŽemontrons qu'elle est auto duale, libre et colibre. Ces rŽesultats sont liŽes au
fait que les permutations de blocs uniformes constituent un monošide inverse
factorisable. Cette alg`ebre de Hopf contient l'alg`ebre de Hopf des permutations
de Malvenuto et Reutenauer et l'alg`ebre de Hopf des fonctions symŽetriques `a
variables non commutatives de Gebhard, Rosas, et Sagan.
1. Uniform block permutations
1.1. Set partitions. Let n be a non-negative integer and let [n] := {1, 2, . . ., n}. A set
partition of [n] is a collection of non-empty disjoint subsets of [n], called blocks, whose
union is [n]. For example, A = {2, 5, 7}{1, 3}{6, 8}{4} , is a set partition of [8] with 4
blocks. We often specify a set partition by listing the blocks from left to right so that

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics