Grounded partitions at level 2 in type A_1^(1): bijections and affine crystal structure

Abstract: Grounded partitions, introduced by Dousse and Konan and motivated by the theory of perfect crystals, are combinatorial objects whose generating functions-associated to specific defining matrices-admit infinite product expressions. In this paper, we investigate the combinatorics of grounded partitions at level $2$ in type $A_1^{(1)}$ with exact difference conditions. We provide the first bijective proof that their generating functions are given by infinite products. Furthermore, we show that these partitions carry an affine crystal structure in type $A_1^{(1)}$. By restricting to the finite type $A$ crystal structure, we obtain new infinite sum expressions for the corresponding products. Remarkably, the major index statistic on Yamanouchi words arises naturally within this framework.