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In settings, Settings > Technical > Sequences & Identifiers > Sequences, for each sequence, one of the options is called Implementation and can be set to either Standard or No Gap. Just based on the name, one could guess the difference but "Standard" is a rather vague term on its own in this context.

What is the formal definition for these two implementation options and how does one choose one or the other?

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I also have to add, that no gap implementation doesn't work. It has been marked as bug in https://bugs.launchpad.net/openobject-server/+bug/1170601 but nothing happened since then.

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No gap means you cannot have a gap in the numbering of documents.
For example, if you create a quotation, it will be numbered as Quotation 001. Then the second you create will be Quotation 002. If you then delete this second quotation and create a third one, it will be:

  • Quotation 002 with "no gap"
  • Quotation 003 with "Standard"

The "No gap" will detect the latest number in use while the standard will keep incrementing no matter what happens to the previous ones.

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I've tested this extensively, and from everything I can see, the "No Gap" implementation does not work. I performed an exact test with purchase order / quote as in your example, and the system continues to use the next sequence number (incremented by one). Please, can something shed some light on this?