Omitting Parentheses from the Cyclic Notation

The purpose of this article is to initiate a combinatorial study of the Bruhat–Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form a bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes.