Rashi on Rosh Hashana 22 omud 2 says that the baytusim wanted the 1st day of sefirah and Shavuos to fall on Sunday because of the possuk which says ממחרת השבת which they interpret literally, that the omer should be brought after shabbos as opposed to after Sunday. What's this got to do with what day Shavuos falls on?
The omer count is 49 days (seven complete weeks) after the bringing of the omer. Thus, Shavuos is one day of the week later than the "shabbos" which begins the omer period. That is day 1, 8, 15, 22, 29, 36, 43, 50 (Shavuos) of the omer are all on the same day of the week.
Since the baisusim force the first day of the omer to be on the day that follows the Shabbat of the chag (rather than after the first day of chag), the "seven complete weeks" will end on Shabbos and Shavuos is forced to be on Sunday. That is why we always have Shavuos on the same day of the week as the second day of Pesach (and the first day of the omer).