Vogan’s conjecture on local Arthur packets ofp-adic GLn and a combinatorial lemma

For GL n over a p-adic field, Cunningham and Ray proved Vogan's conjecture, that is, local Arthur packets are the same as ABV packets.They used endoscopic theory to reduce the general case to a combinatorial lemma for irreducible local Arthur parameters, and their proof implies that one can also prove Vogan's conjecture for p-adic GL n by proving a generalized version of this combinatorial lemma.Riddlesden recently proved this generalized lemma.We give a new proof of it, which has its own interest.