Agarwal and Bressoud (Pacific J. Math. 136(2)(1989) 209–228) defined a class of weighted lattice paths and interpreted several 𝑞-series combinatorially. Using the same class of lattice paths, Agarwal (Utilitas Math. 53(1998) 71–80; ARS Combinatoria 76(2005) 151–160) provided combinatorial interpretations for several more 𝑞-series. In this paper, a new class of weighted lattice paths, which we call associated lattice paths is introduced. It is shown that these new lattice paths can also be used for giving combinatorial meaning to certain 𝑞-series. However, the main advantage of our associated lattice paths is that they provide a graphical representation for partitions with $n+t$ copies of 𝑛 introduced and studied by Agarwal (Partitions with 𝑛 copies of 𝑛, Lecture Notes in Math., No. 1234 (Berlin/New York: Springer-Verlag) (1985) 1–4) and Agarwal and Andrews (J. Combin. Theory A45(1)(1987) 40–49).