The date of Shavuot is directly linked to that of Passover. On Passover, the Jewish people were freed from their enslavement to Pharaoh; on Shavuot they were given the Law and became a nation committed to serving God. Shavuot is celebrated in Israel for one day and in the diaspora (outside of Israel) for two days.
Reform Jews celebrate only one day, even in the diaspora. Karaite Jews and Christians believe that Shavuot always falls on a Sunday, while mainstream Jews follow the teaching of the Talmud, which holds that the holiday commences immediately after the "counting of the omer," or 50 days after Passover. (New World Encyclopedia)
Mainstream Jews and Messianic Jews observe Sivan 6 as Pentecost (fifthieth).
The Torah states that Pentecost is counted from the day after the Sabbath.
Based on the Talmud, mainstream Jews believe that is referring to the day after the Sabbath of Nisan 15 and Pentecost is always on Sivan 6.
Christians believe Pentecost is counted from the day after the weekly Sabbath and Pentecost is always on a Sunday. This is the way the Catholic church counts Pentecost.
He shall wave the sheaf before the LORD, to be accepted on your behalf; on the day after the Sabbath the priest shall wave it. (Leviticus 23:11)