Let $$\chi_S(x,y)=\begin{cases}1&\text{ if }0\leq x<y\leq 1\\0&\text{ else }\end{cases}$$

be the indicator function of the simplex $S=\{(x,y)\in [0,1]^2:x<y\}$. I am interested in an explicit smooth partition of unity. That is, I am looking for smooth positive functions $f_n:\mathbb R^2\to \mathbb R$ so that $\sum_{n=1}^\infty f_n(x,y)=\chi_S(x,y).$

Does anyone have a reference to such an example? I suspect this should be relatively standard but I have been unable to find anything.