The enumeration of own $t$-dimensional subspaces of a space $V_{m}$ over the field $GF(q)$

In the Chevalley algebra over a field $K$ associated with any system of roots, it is allocated the niltriangular subalgebra $N\Phi ( K) $ with the basis $\{e_{r}(r\in \Phi ^{+}) \}$. In 2001 G.P. Egorychev and V.M. Levchuk had been put two problems of a enumeration of ideals: special ideals in the algebras of classical types (the problem 1) and all ideals (the problem 2). At their decision there is the problem of a finding of the number $V_{m,t}, \,1\leq t\leq m$, all own $t$-dimensional subspaces of space $V_{m}$ over the field $GF(q)$. Recently V.P. Krivokolesko and V.M. Levchuk have found an obvious expression for the number $V_{m,t}$ through a multiple sum from $q$-combinatorial numbers. Here by means of the method of coefficients of the calculation of combinatorial sums developed by the author in the late eighties, the integral representation for numbers $V_{m,t}$ is found. As consequence two simple computing formulas for these numbers were received.