If there are 2 men in shul & 1 is within 30 days for his father & the other has yartzeit for his father, who gets to daven at the amud?
The Shloshim takes preference.
However, this is only a precedence, and not a complete deferral, and therefore it is best to split the davening between the two, so that the Shloshim will start, and the Yahrzeit will take over from Ashrei until the end.
This question is debated by authorities.
The Shach (Yoreh De’ah 376:8) cites from the “Minhagim” that the Yahrzeit takes preference, and this is also the ruling given by Rav Gedaliah Schwartz.
However, the Shach concludes that the Shloshim takes precedence, and this is the consensus of authorities (see Biur Halachah 132; Gesher Ha-Chaim Chap. 30).
The option of splitting the davening into two parts is mentioned by the Mishnah Berurah (Biur Halachah 132), and it is particularly fitting for the case of a competing Shloshim and Yahrzeit.