Truncated multinomial and dirichlet-multinoial

I could not find much online (only https://openresearch-repository.anu.edu.au/bitstream/1885/133942/2/b19235343_Barry_Simon_Christopher.pdf).

Is there such a thing as truncated multinomial? Eentually aiming for truncated dirichlet-multinomial?

Maybe not quite what you are looking for but:

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