UPDATE: In the 1.4.3 Beta, TaleWorlds has fixed player-selected formations disappearing after loading the game. Because of this, this mod won't be continued past 1.4.2 live. Thanks for the support!

nota bene: Re-uploaded 1.0.1 as of 12/7/2020 @ 12:20pm PST to ensure it actually has the binary files. I'm not sure why they disappear sometimes, but if you downloaded version 1.0.1 and it wasn't working, try re-downloading.
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This mod generates an external XML file in the mod directory, prefixed with the player character's name, to save and load player party unit formation assignments. 
Formations are saved when the game is saved, and loaded when you load a game, automatically.

This mod creates any folders or files it requires for its path if they do not exist, and inserts no extra data into the savegame itself, meaning that if you remove this mod mid-game, you can still play your game.

The method used is exactly the same as the one I created for Distinguished Service, just without the rest of the mod, in case you don't want extra baggage.

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version 1.0.1: Exactly the same functionality as 1.0.0, except formation files are placed in \Documents\Mount and Blade II Bannerlord\FormationAssignments\ which prevents issues with read/write access to the game's installation directory.
if 1.0.0 works for you, you don't need to update. This is just for people who had issues with that version.
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If you're interested, check out my other mods:

• Distinguished Service, to uplift your most effective soldiers to companion-dom
• Attributes Matter, a highly-configurable and highly-compatible set of gameplay effects that you can assign attributes to
• Sergeants For Hire, for hire-able sergeants and wardens to augment your party size and prisoner capacity