how many links should your AP have?

The Royal Oak and Offshore lines of watches all utilize integrated bracelets, and do not have end links.

Old style clamshell clasp of an Offshore bracelet. Not included in link count.

Audemars Piguet link counts

Royal Oak Offshore
25721BA, 25721ST, 25721TI

Links: 19
Screws: 8
Removable: 5

Royal Oak Offshore
26170BA, 26170ST, 26170TI,
26470BA, 26470ST, 26470TI

Links: 19
Screws: 8
Removable: 5

Royal Oak Offshore
26237ST, 26237TI, 26237BA, 26237OR,

Links: 19
Screws: 10
Removable: 5

Royal Oak 41mm
15500ST, 15500BA, 15500OR,15500PT

Links: 23 (including clasp)
Screws: 11
Removable: 7

Royal Oak 41mm
15510ST, 15510BA, 15510OR, 

Links: 23 (including clasp)
Screws: 10
Removable: 6

Royal Oak Jumbo
15202ST, 16202ST, 15202BA,
16202BA, 15202PT, 16202PT

Links: 22 (including clasp)
Screws: 10
Removable: 6

Royal Oak Chronograph 41
26331ST, 26331BA, 26331OR, 26586IP, 

Links: 23 (including clasp)
Screws: 11
Removable: 7