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Tachanun [and Lamnatzeiach and Keil Erech Apayim] is omitted on Pesach Sheiyni, which is the 14th of Iyar. It is however recited the Mincha prior, on the 13th of Iyar. [Thus when the 13th falls on Shabbos, Tzidkascha is recited by Mincha.] Tachanun is likewise omitted on the night of the 14th in Kerias Shema Sheal Hamita. It is questionable whether it is to be recited on Motzei Pesach Sheiyni, which is the night of the 15th. [However, on the day of the 15th Tachanun is recited.]
 Siddur Admur; Siddur Shlah; Elya Raba 131:14; Shalmei Tzibur p. 151; Siddur Yaavetz; Shaareiy Efraim 10:27; Shaareiy Teshuvah 131:19; Ashel Avraham Butchach 131; Divreiy Nechemiah 131; Kaf Hachaim 131; Not listed in Shulchan Aruch Chapter 131. See Aruch Hashulchan 131:12 and Yalkut Avraham 493 who wonders at this matter.
Other Opinions: Some Poskim rule that Tachanun is to be recited on Pesach Sheiyni. [P”M 131 M”Z 14; Noam Megadim 26; custom of Chazon Ish; This was the widespread Ashkenazi custom] It is interesting to note that one of the main claims of the Misnagdim against the Chassidim in the Cherem of 1772 was that they did not say Tachanun on Pesach Sheiyni, and held feasts on the day even when it fell on Behab. [Nitei Gavriel Pesach 3 57:3]
 Piskeiy Hassidur 151 and 179; Sichas Kodesh 6th Tishrei 1975 based on Admur in Siddur that Lamnatzeiach is never recited when Tachanun is omitted. This ruling in the Siddur is based on Kneses Hagedola 131:3; Peri Chadash 131:1; Mamar Mordechai 131:6; Kaf Hachaim 581:78 states that so is the custom of the Sefaradim.
Ruling of Admur in the SHU”A: In the Shulchan Aruch, Admur rules like Rama 131:1 that Lamnatzeiach may be recited even on a day that Tachanun is omitted. [Admur 429:12; 602:4 [in parentheses]]
 Siddur Tehillas Hashem [from 1978 and onwards with the Rebbe’s approval] based on Sichas Kodesh 6th Tishrei 1975 that the Rebbe equates Kel Erech Apayim with Lamnatzeiach. See Shulchan Menachem 3:293; Glosses of Rav Raskin p. 165; Hiskashrus 438 p. 17; To note however that the Rebbe himself did recite Kel Erech Apayim even when Tachanun was not recited. [Rav Raskin ibid]
Ruling of Admur: In the Shulchan Aruch [429:12; 602:4] Admur rules that Kel Erech Apayim is recited with exception for Erev Pesach. This follows the ruling of the Hagahos Maimanis; Darkei Moshe 602 brought in Kaf Hachaim 602:12. [Some explain that it is for this reason that Admur in 602:4 placed this ruling in parentheses, as it is not brought in previous Poskim in Shulchan Aruch. [Shulchan Menachem ibid footnote 2] Vetzaruch Iyun as it is brought by Admur in 429:12 without parentheses, and is likewise clearly implied from the Rama 429:2] To note that also in the Siddur before Kel Erech Apayim Admur lists a number of days that it is to be omitted on, thus implying it is not always omitted when Tachanun is not said.
Ruling and explanation of Divrei Nechemia: The Divrei Nechemia 131:9 brings three opinions regarding the saying of Keil Erech Apayim: 1) It is only omitted by a day that Halel is recited or a day that has the status of a Yom Tov like Erev Pesach. [This is the opinion of the Rama 429:2] 2) It is omitted on every Erev Yom Tov and on Isru Chag. 3) Every day that Tachanun is omitted so is Lamnatzeiach. [so rules Peri Chadash 131] The Divrei Nechemia concludes that by Lamnatzeiach we rule like the third opinion, however by Keil Erech Apayim we rule like the opinion that rules it is to be recited.
The reason behind the Rebbe’s ruling: The Rebbe explains that the ruling in the Shulchan Aruch of Admur follows the ruling of the Rama that differentiates between the laws of Tachanun and that of Lamnatzeiach. However according to Admur in the Siddur that rules Lamnatzeiach is always omitted when Tachanun is not said then likewise Kel Erech Apayim is to be omitted in all days that Tachanun is not said. [Sichas Kodesh ibid, printed in Shulchan Menachem ibid] As for the reason why Kel Erech Apayim is omitted this is because it mentions sin, and we do not desire to mention sin on any day that Tachanun is omitted. [See Shaar Hakolel 36:2; Shulchan Menachem ibid footnote 5]
 Siddur Admur; Divreiy Nehemiah 131; Hayom Yom 13th Iyar
Other Opinions: Some Poskim rule that Tachanun is omitted also from the Mincha prior. [Shaar Yissachar Pesach Zeira in name of Arizal; Nitei Gavriel 57:4 in name of Sefarim; Siddur Arizal Chemdas Yisrael]
 Hiskashrus 1033
 Sichas 1983 Pesach Sheiyni letter 12; Behar Bechukosaiy 1983 letter 42; See Otzer Minhagei Chabad Iyar 42
 Machazik Bracha 131:4; Beis David 53; Shaareiy Teshuvah 131:19; Rebbe ibid based on Siddur Admur
Other opinions: Some are accustomed not to say Tachanun also on the 15th. [Shalmei Tzibur p. 151; See Otzer Minhagei Chabad p. 267; Regarding the opinion of the Munkatcher, in Darkei Chaim Veshalom 191 it states that he did not say Tachanun on the 15th and in Nimkei Orach Chaim 131:6 and in Shaar Yissachar [Pesach Zeira] he writes that according to the students of the Arizal it is to be omitted. The Rebbe ibid however explains that most likely the Munkatcher held Tachanun is omitted by the night of the 15th and not the day.]
Custom to omit Tachanun for seven days: Some are accustomed to omit Tachanun for seven days until the Tashlumin of Pesach Sheiyni expires. [Likkutei Maharich in name of Hayashar Vehatov of Liska; Custom of Divrei Chaim and Tzanz; See Piskeiy Teshuvos 492:1 footnote 8] Others negate this custom. [Darkei Chaim Veshalom 191; Nimkei Orach Chaim 131:6; Shaar Yissachar [Pesach Zeira]]