Abstract. Recently, the first author has studied hook length formulas for partitions in a systematic manner. In the present paper we show that most of those hook length formulas can be generalized and include more variables via the Littlewood decomposition, which maps each partition to its t-core and t-quotient. In the case t = 2 we obtain new formulas by combining the hook lengths and BG-ranks introduced by Berkovich and Garvan. As applications, we list several multivariable generalizations of classical and new hook length formulas, including the Nekrasov-Okounkov, the Han-Carde-LoubertPotechin-Sanborn, the Bessenrodt-Bacher-Manivel, the Okada-Panova and the Stanley-Panova formulas.

Andrews, Lewis and Lovejoy introduced the partition function P D(n) as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that P D(3n + 2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of P D(3n + 2) which implies the congruence of Andrews, Lewis and Lovejoy. For P D(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of (1 − q 2n+1 ) times a function F (q 2 ). We find an explicit formula for F (q 2 ), which leads to a formula for the generating function of P D(3n). We also obtain a formula for the generating function of P D(3n + 1). Our proofs rely on Chan's identity on Ramanujan's cubic continued fraction and some identities on cubic theta functions. By introducing a rank for the partitions with designed summands, we give a combinatorial interpretation of the congruence of Andrews, Lewis and Lovejoy.

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We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.