R.T. Concerning Sephardim, Tachanun is not said for the remainder of Tishrei after Shmini Atzeret (Yemei Tashlumim for Shalmei Chagiga) and for the 6 or 7 days following Shavuos (Same reason).
What RT mentioned for the days following Shavu’os is correct [albeit 5-6 days]; However following Shemini Atzeres, there is no Tashlumin, since Sukkos was already 7 days [unlike Shovu’os only being 1 day] and Shemini Atzeres is just an extra day/chance to make up Chaggigo of Sukkos tagged onto the 7 days.
Reason why those who omit Tachnun in last week of Tishri, is because since majority of month already did not have Tachnun, so follow by majority of month.
To clarify:- Most Hassidic places don’t say Tachnun for week after Shovu’os & Shemini Atzeres, whereas most Litvac places do say already 2 days after Shovu’os & Simchas Torah, and Yekke places say 2 days after Shovu’os, but not last week of Tishri.
Getting back to original issue raised by abhaven, Tachnun is universally omitted on the following Yorzeits:- Aharon haKohen [Ov 1st]; Miriam haNevi’oh [Nisson 10th]; Nodov va’Avihu [Nisson 1st]; Yehoshu’a bin Nun [Nisson 26th]; hoRav Avigdor haKohen Miller [Nisson 27th]; Vilna Gaon [Tishri 19th]; Shmuel haNovi [Yom Yerusholayim]; hoRav Shimon Eider [Tishri 16th]; However it is recited on Moshe Rabeinu’s Yorzeit. By the way, Tachnun is also omitted on birthday of Yitzchok Ovinu [Nisson 15th:- Exactly 400 years prior to Yetzi’as mitzraim to the very day]