On a generalized 6-way combinatorial identity

In this paper, we study a generalized q-series combinatorially using four different combinatorial tools, viz., associated lattice paths, generalized F-partitions, anti-hook differences, and Bender-Knuth matrices. Here, the new results provide extension to our recent work, ‘On a generalized basic series and Rogers–Ramanujan type identities’, Discrete Math. 18(1) (2023) 15–28. In this paper, we successfully establish bijections between the six diverse combinatorial classes that results in a 6-way combinatorial identity. This work provides a precise way to establish inter-relations among different combinatorial objects which leads to better categorification of combinatorial identities. In this process, we obtain several 7-way Rogers–Ramanujan–MacMahon type identities as particular cases.