We introduce and investigate a family of MacMahon-type q-series \( H_k^{\pm }(q)=\sum _{n=0}^{\infty } h_k^{\pm }(n)q^n=\sum _{1 \le n_1 \le \dots \le n_k} \prod _{i=1}^k \frac{q^{n_i}}{1 \mp q^{n_i}}, \) 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 \(H_{k,m}^{\pm }(q)\) and their limiting forms \(H_k^{\pm }(q)\) . In particular, we derive inversion formulas expressing \(H_k^{\pm }(q)\) in terms of the q-products \((\pm q;q)_\infty \, q^k/(q;q)_k\) . These identities lead to new combinatorial interpretations for the coefficients of \((\pm q;q)_\infty \, q^k/(q;q)_k\) in terms of signed counts of partitions, overpartitions and partition pairs. Several applications to explicit formulas for the partition functions \(h_k^{\pm }(n)\) are also obtained.
