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We study critical percolation on a regular planar lattice. Let $E_G(n)$ bethe expected number of open clusters intersecting or hitting the line segment$[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrictto the upper halfplane, or $\mathbb{C}$, when we consider the full lattice).Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that$E_{\mathbb{H}}(n) = An + \frac{\sqrt{3}}{4\pi}\log(n) + o(\log(n))$, where $A$is some constant. Recently Kov\'{a}cs, Igl\'{o}i and Cardy (2012) derivedheuristically (as a special case of a more general formula) that a similarresult holds for $E_{\mathbb{C}}(n)$ with the constant $\frac{\sqrt{3}}{4\pi}$replaced by $\frac{5\sqrt{3}}{32\pi}$. In this paper we give, for sitepercolation on the triangular lattice, a rigorous proof for the formula of$E_{\mathbb{H}}(n)$ above, and a rigorous upper bound for the prefactor of thelogarithm in the formula of $E_{\mathbb{C}}(n)$.

The expected number of critical percolation clusters intersecting a line segment