Using standard double-coset formulas to obtain general-n results for Sylow double cosets in Sn

Develop methods that employ the orbit–stabilizer formula (2.4) and the standard double-coset enumeration formulas (2.5a), (2.5b), and (2.5c) to derive general-n results about the number and sizes of double cosets in Pn \ Sn / Pn, beyond example-driven computations.

Background

The paper reviews standard formulas for counting double cosets, including the orbit–stabilizer approach and character-theoretic and class-sum formulas. These tools are classical and, in principle, applicable to H \ G / K problems.

However, the authors report that, despite these formulas being useful for preliminary computations and examples, they did not lead to general results in the setting of Pn \ Sn / Pn, indicating a methodological gap that invites further development.