Enumeration of Up-Down Permutations by Upper Records.
Dans cet article nous caractérisons, par les facettes, l’enveloppe convexe des vecteurs caractéristiques des hyperplans d’un espace projectif fini et d’un espace affine fini.
Dans cet article nous caractérisons, par les facettes, l'enveloppe convexe des vecteurs caractéristiques des hyperplans d'un espace projectif fini et d'un espace affine fini.
Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...
Let denote the isometry group of . We prove that if G is a paradoxical subgroup of then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system of Jordan domains with differentiable boundaries and of the same volume such that has the cardinality of the continuum, and for every amenable subgroup G of , the elements of are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...