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All men are obligated in the Mitzvah of Sefiras HaOmer.
Is an Onen obligated to recite Sefiras Haomer?
An Onen is exempt from Sefiras Haomer until after the burial. Nevertheless, if not counting will cause him to lose the ability to say a blessing on the coming nights [i.e. he is an Onen for a couple of days], he is to count without a blessing during his Aninus, and may then continue to count with a blessing on the later nights after the burial.
If the burial will take place during the day: In the event that one became an Onen at night, prior to counting Sefira, and the burial will only be taking place the next day, some Poskim rule he is not to count at night, but is rather to count the next day, after the burial, without a blessing. [Seemingly however, in such a situation, it is best to count at night without a blessing, and then count again the next day, after the burial, without a blessing. ]
Women are exempt from the Mitzvah of Sefiras HaOmer. [Likewise, based on Kabala, women are not able to fulfill this Mitzvah.] However in certain provinces women have accepted this Mitzvah upon themselves as a complete obligation. [In such provinces, the women are obligated to count due to this custom.] [Practically, the Chabad custom is for women to recite Sefiras HaOmer. Likewise, due to other reasons, all women should recite Sefiras Haomer in the beginning of the night even if they have missed a day and can no longer say a blessing, as it is possible that they must refrain from work throughout the entire night until they do so. The husbands should remind their wives to recite the Sefira each night.]
Women are to count Sefiras Haomer each night, in the beginning of the night, and are to be reminded by their husbands to do so. This applies even if they already missed a day. If they have not missed a day, they may count at night with a blessing.
Children who have reached the age of Chinuch are to be educated in the Mitzvah of Sefiras HaOmer. They are to recite it at night with a blessing [if they have not missed a previous night], following the same laws as adult men. The age of Chinuch is from the age that the child understands the meaning of Sefiras Haomer.
In areas that night begins very late, must one keep the child awake until late at night in order to say Sefiras Haomer?
It is proper to wake the child up after night in order to say Sefira. This applies only if the child has reached the age of Chinuch. If this poses difficulty, one may simply recite Sefira with the child during the day.
 489/1 “Every single Jew”
Slaves: Slaves are exempt from the Mitzvah just like women. [Admur 489/2]
 Noda Beyehuda O.C. 1/27, brought in Pischeiy Teshuvah 341/6; Gilyon Maharsha 341; Rav Akiva Eiger 489; Biur Halacha 489/8 “Belo Bracha”; Kaf Hachaim 489/86; Chelkas Yaakov 3/65; Piskeiy Teshuvos 489/23; See Divrei Moshe 1/29 that the Saraf of Magalintza asked someone to count on his behalf while he stood and listened without intent to perform the Mitzvah.
Other opinions: Some Poskim rule that one cannot count Sefira during Aninus even if it causes him to lose the blessing for the remainder of Sefira. [Nehar Shalom, brought in Biur Halacha ibid]
 See Admur 71/1; Nehar Shalom, brought in Biur Halacha ibid; See however Noda Beyehuda ibid who argues that perhaps an Onen is obligated in Sefira, as it does not take very long to say and does not distract him from the Mitzvah of burial. He uses this argument to allow saying it without a blessing in the event that he may lose the blessing all together for the coming nights.
 Biur Halacha ibid in name of Nehar Shalom
 See Biur Halacha ibid that perhaps the day counting according to the opinion who allows counting by day, is a mere Tashlumin. Accordingly, it won’t help for him to count by day, the next day, even according to their opinion, as if he was exempt at night then he will also be exempt the next day even after the burial. [See Admur 71/1 and Michaber 341/2] Therefore, he should count at night in order not to forfeit the blessing, and then count again the next day after the burial.
 489/2; M”A 489/1; See Piskeiy Teshuvos 489/7
Other opinions: Some Poskim hold that the Mitzvah of Sefiras Haomer is not considered a time dependent command, and women are hence obligated in the Mitzvah. [Ramban Kiddushin 34a; See Avnei Nezer 384; See Divrei Malkiel 3/5 that this is a misprint in Ramban] The reason for this is because the Mitzvah is dependent on the Omer sacrifice and not on a specific time. [Shitas Hakamonim Baba Kama p. 334; Kinyan Torah 4/44 based on Turei Even Megillah 20b] Alternatively, it is because women are obligated in Matzah and Chametz, and are hence also obligated in the Mitzvah of Sefiras Haomer, which begins during Pesach. [Avnei Nezer ibid] See also Mikraeiy Kodesh 2/67 and Piskeiy Teshuvos ibid
 Admur ibid; M”A ibid; Rambam Temidim Umusafim 7/24; Chinuch 306
The reason: As this Mitzvah is a time dependant command, of which women are exempt. [Admur ibid] This is learned from the verse “Lemaan Tihiyeh Toras Hashem Bificha” that is written regarding Tefillin which teaches that the entire Torah is connected to the Mitzvah of Tefillin. Now just like women and slaves are exempt from wearing Tefillin, due to the reason explained in 38/3, and Tefillin is a time dependent Mitzvah, as the Mitzvah of Tefillin does not apply on Shabbos and Yom Tov, so too from here we learn that all time dependent Mitzvos women and slaves are exempt from, with exception to some specific Mitzvos. [Admur 17/1; Levush 17/2]
 M”A ibid based on Zohar Tetzaveh and Emor p. 97b “The 50 days are days of the male world, and are given only to men”; brought also in Birkeiy Yosef 489/22; Daas Torah 489/1; Avnei Nezer 384; Kaf Hachaim 489/9; Rav Poalim 1/12 based on Kabbalah
 So writes Admur ibid; However see M”A ibid who writes “Although women are exempt, the women have already accepted the Mitzvah upon themselves as an obligation.” He does not distinguish between provinces, and certainly does not limit it to certain provinces as writes Admur ibid. The M”B 489/3 writes that in these provinces women are not accustomed to count Sefira at all. In the home of the Steplier women would count Sefiras Haomer. [Orchos Rabbeinu 2/92]
 Admur ibid; M”A ibid; Chok Yaakov 489/3; Elya Raba 489/2
Other opinions: Some Poskim question the above ruling, stating that there is no source for saying that women accepted this Mitzvah upon themselves as an obligation. [Minchas Chinuch Mitzvah 306]
 See regarding Shofar and Lulav: Admur 17/3; Shut Rav Akiva Eiger 1 Hashmatos; Ashel Avraham Butchach 689 and Mahadurah Tinyana 592
 Hiskashrus; Otzer Minhagei Chabad Nissan p. 254
 See Admur 493/9; brought in Halacha 5!
 See Piskeiy Teshuvos ibid footnote 35
 Admur ibid
Other opinions: Some Poskim rule that women are not to count with a blessing. [Shulchan Shlomo brought in M”B 489/3; Kaf Hachaim 489/9] As they will certainly miss a certain night and hence even initially we know their counting is incomplete. [Shulchan Shlomo ibid] To note however that according to the Rebbe’s explanation of the dispute in 489/23 that in truth all agree that each day is a separate Mitzvah, and missing a day only holds the future days in question, there is no worry whatsoever if the women miss a future day as it has no affect, according to any opinion on the previous Mitzvos fulfilled. [Se Likkutei Sichos 1/270]
Other opinions and Sefaradi custom: According to the Sefaradic custom, women are not to say the blessing. [Kaf Hachaim ibid] In addition to the above-mentioned issue, some Poskim rule that women may never recite a blessing over any Mitzvah that they are not obligated in. [Michaber 589/6; Beis Yosef 589; Rambam Hilchos Tzitzis; Igur in name of Reb Yeshaya; Rashi, brought in Hagahos on Rambam] The Michaber ibid rules that they cannot say a blessing as Safek Brachos Lihakel. [Beis Yosef 589] The Birkeiy Yosef ibid states that after seeing the answer from heaven in Shut Min hashamayim he retracted his ruling like the Michaber and would tell women to recite a blessing. He concludes that if the Michaber would have seen the Shut Min Hashamayim certainly he too would have retracted his ruling.
 The reason: There is no worry of prohibition involved for women to recite a blessing on Sefiras Haomer despite the fact that they are exempt, as explained in 17/3. [Admur ibid] The reason for this is because men are commanded in this Mitzvah and when a woman performs a command that she is exempt from fulfilling, although she is not rewarded as much as one who was commanded, nevertheless she does receive some reward. For this reason they are permitted to recite a blessing over all the Mitzvos that they are exempt from, if the Mitzvah is performed properly, in a way that a man would fulfill his obligation. [Admur 17/3]
 Admur 343/2-3 regarding all Mitzvos, and in accordance to 1st opinion in Admur 70/2; Shaareiy Teshuvah 489/20 “Our custom is that children who have reached the age of Chinuch count with a blessing”; Maharil Hilchos Sefiras Haomer; Regarding Mitzvos that apply in early morning or at night, during hours a child might not be awake-see Admur 70/2 regarding Shema that this matter is disputed in Poskim although concludes and that one should follow the first, stringent, opinion; M”B 426/1 that states this dispute regarding Shema would likewise apply regarding Kiddush Levana, and accordingly, the same should apply regarding Sefiras Haomer. To note, that according to those who rule that the Mitzvah of Sefira applies also by day, this dispute is irrelevant, and one is obligated to educate his child in Sefira according to all.
 As is the law by an adult; However see Nitei Gavriel 24/7 footnote 13 who novelizes that they may continue to count with a blessing.
 Admur 343/3
 See Admur 70/2 regarding Shema