how many links should your patek have?

The Nautilus line of watches utilize integrated bracelets, and do not have end links.

This is a 5905/1A, with endlinks and a “hidden clasp” buckle.

Patek philippe link counts

Nautilus
5711, 5712, 5980, 5990, 5740

Links: 19
Screws: 8
Removable: 5

Annual Calendar
5905/1A

Links: 14
Pins: 10 (2 on clasp)
Removable:

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