## List of issues > Series «Mathematics». 2016. Vol. 17

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The Enumeration of own t-Dimensional Subspaces of a Space Vm 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 NΦ(K) with the basis {e_{r}(r ∈ Φ^{+})}. 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 ≤ t ≤ 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.

**MSC**

05+20

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