How many cable managers should I plan for per patch panel?

No. There is no useful one-manager-per-patch-panel rule.

A lightly populated panel with short routes may need very little horizontal management; a dense installation with cables travelling to both sides of the rack may justify considerably more.

Switch position, patch-lead length and any vertical management all form part of the same system.

Counting panels and buying the same number of managers is temptingly simple, but rack cabling rarely rewards that sort of arithmetic.