Natural labeling of size p^2 double cosets for Cp in Sp

Construct a natural labeling of the double cosets of size p^2 in Pp \ Sp / Pp, where Pp is a Sylow p-subgroup of Sp (isomorphic to the cyclic group Cp), providing an explicit and canonical parametrization of these double cosets.

Background

In the smallest nontrivial case n=p, the Sylow p-subgroup Pp of Sp is cyclic of order p, and the double cosets Pp w Pp have sizes p or p^2. Minimal-size double cosets are naturally labeled by NG(Pp)/Pp via standard arguments.

Despite this, the authors remark that even in this basic setting there is no known natural labeling for the double cosets of maximal size p^2, highlighting a concrete gap in understanding the structure of these double cosets.