Abstract We introduce and investigate a family of MacMahon-type q -series Hₖ^ (q) = ₍=₀^ hₖ^ (n) qⁿ= ₁ ₍䃑 ₍䂵 ₈=₁ᵏ q^nᵢ1 q^{nᵢ}, H k ± (q) = ∑ n = 0 ∞ h k ± (n) q n = ∑ 1 ≤ n 1 ≤ ⋯ ≤ n k ∏ i = 1 k q n i 1 ∓ q n i, which enumerate weighted partitions according to a fixed number of (not necessarily distinct) magnitudes. These series extend the classical generating functions for partitions with a prescribed number of distinct parts originally studied by MacMahon. Using Gaussian polynomials, we establish finite and infinite linear relations between the truncated series H₊, ₌^ (q) H k, m ± (q) and their limiting forms Hₖ^ (q) H k ± (q). In particular, we derive inversion formulas expressing Hₖ^ (q) H k ± (q) in terms of the q -products (q;q) _ \, qᵏ/ (q;q) ₖ (± q ; q) ∞ q k / (q ; q) k. These identities lead to new combinatorial interpretations for the coefficients of
Mircea Merca (Thu,) studied this question.