"Styczeń ma trzydzieści jeden dni."

Translation: January has thirty-one days.

[eannaoc]

Is "dni" here in accusative or genetive? Does the 'jeden' alter the case here (?accus) or the trzydzieści (?gen)?

[Jellei]

Very good question. It's Genitive, though.

Compare "trzydzieści jeden kobiet" (Genitive) and "trzydzieści dwie/trzy/cztery kobiety" (Accusative) to see it more easily.