The object to be considered is a kind of super spline spaces with homogeneous boundary conditions \[ S_ d^{r, p,\alpha} (\Delta)= \{s\in C^ \alpha (\mathbb{R}^ 2)\cap S_ d^{r,\rho} (\Delta):\;s|_{\mathbb{R}^ 2 \setminus\Omega} \equiv 0\}, \] where \(\Omega\subset \mathbb{R}^ 2\) is a simpler-connected Jordan polygonal region, \(\Delta\) is an arbitrary regular triangular cut of \(\Omega\), \(V\) the number of its vertex \(\nu_ 1, \nu_ 2,\dots, \nu_ V\). For given positive integers \(d\), \(r\), \(\rho\), \(r\leq \rho<d\), a spline space of degree \(d\) and order of smoothness \(r\) is \(S_ d^ r (\Delta)= \{s\in C^ r (\Omega)\): the limitation of \(s\) on any triangle \(\in\pi_ d\}\), a super spline space of degree \(d\) and order of smoothness \(r\), \(\rho\) is \(S_ d^{r,\rho} (\Delta)= \{s\in S_ d^ r (\Delta)\): \(s\in C^ \rho (\nu_ i)\), \(i=1,\dots, V\}\), \(\pi_ d\) denotes the set of bivariate polynomials of degree \(\leq d\). In this paper a dimension formula for \(S_ d^{r,\rho,d} (\Delta)\) is obtained under the conditions \(\rho\geq 2r\), \(d\geq 2\rho+1\), \(0\leq \alpha\leq r\) as well as \(r\leq\rho\leq 2r\), \(d\geq 4r+1\), \(0\leq\alpha \leq \rho-r\) respectively.