A solution to one of Knuth's permutation problems

We answer a problem posed recently by Knuth: an n-dimensional box, with edges lying on the positive coordinate axes and generic edge lengths W_1 < W_2 < ... < W_n, is dissected into n! pieces along the planes x_i = x_j. We describe which pieces have the same volume, and show that there are C_n distinct volumes, where C_n denotes the nth Catalan number.Comment: 4 pages, 2 figures

The isometries of the cut, metric and hypermetric cones

We show that the symmetry groups of the cut cone Cut(n) and the metric cone Met(n) both consist of the isometries induced by the permutations on {1,...,n}; that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4). This is then extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone.Comment: 8 pages, LaTeX, 2 postscript figure