Buchstaber invariants of skeleta of a simplex

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A moment-angle complex Zₖ is a compact topological space associated with a finite simplicial complex K. It is realized as a subspace of a polydisk (D²)ᵐ, where m is the number of vertices in K and D² is the unit disk of the complex numbers, and the natural action of a torus (S¹)ᵐ on (D²)ᵐ leaves Zₖ invariant. The Buchstaber invariant s(K) of K is the maximum integer for which there is a subtorus of rank s(K) acting on Zₖ freely. The story above goes over the real numbers in place of and a real analogue of the Buchstaber invariant, denoted s(K), can be defined for K and s(K) s(K). In this paper we will make some computations of s(K) when K is a skeleton of a simplex. We take two approaches to find s(K) and the latter one turns out to be a problem of integer linear programming and of independent interest.