The rook monoid $R_n$ is the finite monoid whose elements are the 0-1 matrices with at most one nonzero entry in each row and column. The group of invertible elements of $R_n$ is isomorphic to the symmetric group $S_n$. The natural extension to $R_n$
of the Bruhat-Chevalley ordering on the symmetric group is defined in cite{Renner86}. In this paper, we find an efficient, combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also give a useful, combinatorial formula for the length function on $R_n$.
In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $TT^2$-equivariant locally free sheaf on $Hil{n,n-1}$. Applying
our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients . We call this conjecturally positive polynomial as textsl{the nested $q,t$-Cat alan series}, for it has many conjectural properties similar to that of the $q,t $-Catalan series.
