Component-band vertex count ($N_{[k,\ell )}(G)={\sum }_{C:k\le |C|<\ell }|C|$)

This is the total number of vertices in graph components with orders in the specified half-open interval. It differs from the number of those graph components. Write $N_{\ge k}$ for the total above the lower endpoint. Then $N_{[k,\ell )}=N_{\ge k}-N_{\ge \ell }$. Disjoint size bands count disjoint sets of vertices at a common time, even though their mass may move between bands as an Achlioptas process adds edges.