Smirnov words and the Delta Conjectures

We provide a combinatorial interpretation of the symmetric function ${{\mathrm{\Theta }}_{e_k}{\mathrm{\Theta }}_{e_l}\mathrm{\nabla }{e}_{n-k-l}\mid }_{t=0}$ in terms of segmented Smirnov words. The motivation for this work is the study of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, whose Frobenius characteristic is conjectured to be the symmetric function in question. Furthermore, this function is related to the Delta conjectures. Our work is a step towards a unified formulation of the two versions, as we prove a unified Delta theorem at $t=0$.