$A_1^{(1)}$-Grounded partitions at levels $1$ and $2$, Part I: bijections

Grounded partitions, introduced by Dousse and Konan, are coloured partitions satisfying difference conditions encoded by a matrix.

For suitable choices of this matrix, their generating functions are known to coincide with characters of affine Lie algebras.

In this paper, we study, from a combinatorial point of view, the grounded partitions introduced by Dousse, Hardiman and Konan and related to the Lie algebra $A_1^{(1)}$.

Using the connection with characters, they showed that the generating function for these grounded partitions is an infinite product.

We give direct combinatorial proofs of the corresponding product formulas.

In particular, we construct two explicit bijections from grounded partitions to odd overpartitions, and to partitions in which the even parts are distinct.