The Tur (R. Yaakob Ben Asher, 1343-1269, Germany-Spain) in Siman 428 brings down that the Jewish Calendar is arranged so that the first day of Rosh Hashana can never fall on a Sunday, Wednesday or Friday. This is represented by the mnemonic phrase, "Lo A'D'U' (numerically equivalent to 1,4,6) Rosh," meaning Rosh HaShana will not fall on the first, fourth or sixth day.
The reason for this is as follows: If Rosh HaShana would fall on Sunday, Hoshana Rabba would fall on Shabbat, and the Misva of the Araba would be able to be performed. If Rosh Hashana would fall on Wednesday, Yom Kippur would be on Friday. This would create a situation of two consecutive days prohibited from the Torah to do Melacha. If someone, Heaven forbid, would pass away on Yom Kippur, he could not be buried until Sunday, and Heaven forbid, the body might begin to decompose. Similarly, Rosh Hashana cannot fall on Friday, to avoid the reverse situation in which Yom Kippur would be on Sunday, and the same problem of disrespect to the dead would exist.
Parashat Ki Tavo is always read before Rosh Hashana, so that the curses detailed in it are on last year's account. The following Parasha, Nisavim, is read on the Shabbat immediately prior to Rosh Hashana. Some years Nisavim is read by itself and other years it is read together with the following Parasha, VaYelech. The Shulhan Aruch presents a mnemonic phrase, "BaG (numerically equivalent to 2,3=Monday, Tuesday) Hamelech, Pat VaYelech"-If the Melech (King), which alludes to Rosh Hashana, is on Monday or Tuesday, split Parashat VaYelech from Nisavim. However, if Rosh Hashana is on any other day, they are combined. This year, Rosh Hashana starts on Monday, and therefore Nisavim is read by itself.
Rosh Hashana never falls on Sunday, Wednesday or Friday.