Congruences for hook lengths of partitions

Recently, Amdeberhan et al. [Proc. Amer. Math. Soc. Ser. B 11 (2024), pp. 345–357] proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer n n , thus answering positively a conjecture raised by Ballantine et al [Res. Math. Sci. 10 (2023), pp. 36]. In this paper, we show how these congruences can be immediately derived and generalized from an addition theorem for self-conjugate partitions proved by the second author. We also recall how the addition theorem proved before by Han and Ji [Trans. Amer. Math. Soc. 363 (2011), pp. 1041–1060] can be used to derive similar congruences for the whole set of partitions, which are originally due to Bessenrodt [Ann. Comb. 2 (1998), pp. 103–110], and Bacher and Manivel [Sém. Lothar. Combin. 47 (2001/02), pp. 11]. Finally, we extend such congruences to the set of z z -asymmetric partitions defined by Ayyer and Kumari [J. Algebra 609 (2022), pp. 437–483], by proving an addition-multiplication theorem for these partitions. Among other things, this contains as special cases the congruences for the number of hook lengths for the self-conjugate and the so-called doubled distinct partitions.