On the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex

We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions, and find one unique class of them satisfying the symmetric interlacing property. This leads us to obtain an explicit formula for a class of 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex through establishing its connection with 1-leg Donaldson-Thomas $\mathbb{Z}_4$-vertex using the vertex operator methods of Okounkov-Reshetikhin-Vafa and Bryan-Young.