The Eulerian Distribution on the Fixed-Point Free Involutions of the Hyperoctahedral Group Under the Natural Order

Two totally order relations are defined on the hyperoctahedral group \({\mathfrak {B}}_n\). Regarding \({{\mathfrak {B}}}_n\) as a Coxeter group of type B, we always use the natural order. By taking \({{\mathfrak {B}}}_n\) as a color permutation group, it is convenient to use the r-order. Considering \({{\mathfrak {B}}}_n\) as a colored permutation group, Moustakas proved that the Eulerian distribution on the involutions of \({{\mathfrak {B}}}_n\) is unimodal, Cao and Liu proved that it is \(\gamma \)-positive, they also proved that the Eulerian distribution on the fixed-point free involutions of \({{\mathfrak {B}}}_n\) is symmetric, unimodal and \(\gamma \)-positive. In this paper, we prove that the Eulerian distribution on the fixed-point free involutions of \({{\mathfrak {B}}}_n\) is symmetric, unimodal and \(\gamma \)-positive when \({{\mathfrak {B}}}_n\) is viewed as Coxeter group of type B.