On the spectrum of the number of geodesics and tight geodesics in the curve complex

Let $S$ be an oriented surface of type $(g,n)$. We are interested in geodesics in the curve complex $\mathcal{C}(S)$ of $S$. In general, two $0$-simplexes in $\mathcal{C}(S)$ have infinitely many geodesics connecting the two simplexes while another geodesics called tight geodesics are always finitely many. On the other hand, we may find two $0$-simplexes in $\mathcal{C}(S)$ so that they have only finitely many geodesics between them. In this paper, we consider the spectrum of the number of geodesics with length $d(\ge 2)$ in $\mathcal{C}(S)$ and tight geodesics, which is denoted by ${\mathfrak{S}\mathfrak{p}}_d(S)$ and ${\mathfrak{S}\mathfrak{p}}_d^T(S)$, respectively. In our main theorem, it is shown that ${\mathfrak{S}\mathfrak{p}}_d(S)\subset {\mathfrak{S}\mathfrak{p}}_d^T(S)$ in general, but ${\mathfrak{S}\mathfrak{p}}_2(S)={\mathfrak{S}\mathfrak{p}}_2^T(S)$. Moreover, we show that ${\mathfrak{S}\mathfrak{p}}_2(S)$ and ${\mathfrak{S}\mathfrak{p}}_2^T(g,n)$ are completely determined in terms of $(g,n)$.