Combinatorial point configurations and polytopes

The monograph is dedicated to exploring combinatorial point configurations derived from mapping a set of combinatorial configurations into Euclidean space. Various methods for this mapping, along with the typology and properties of the resultant configurations, are presented. In addition, the study revolves around combinatorial polytopes defined as convex hulls of combinatorial point configurations. The primary focus lies in examining multipermutation and partial multipermutation point configurations alongside their associated combinatorial polytopes known as multipermutohedra and partial multipermutohedra. Our theoretical contributions are substantiated through the proof of theorems and supporting auxiliary statements. Examples and illustrations are included to enhance the comprehension of the material.

Introduction 9 1 Background of finite point configurations 15 1.1 Sets and multisets, order relations and mappings 15 1.2 Finite point configurations and their decompositions 20 1.3 Convex hulls of finite point configurations 29 1.4 Functional representations of finite point configurations 35 2 Finite and combinatorial point configurations 45 2.1 Properties of finite point configurations 45 2.2 Multilevel finite point configurations and multilevel polytopes 51 2.3 Operations on finite point configurations 58 2.3.1 Subsets of finite point configurations 61 2.3.2 Intersection of finite point configurations 63 2.3.3 Intersection of finite point configurations and surfaces 65 2.3.4 Intersection of finite point configurations and strictly convex surfaces 66 2.3.5 Union of finite point configurations 67 2.3.6 Minkowski sum and difference of finite point configurations 69 2.3.7 Hadamard product of finite point configurations 71 2.3.8 Direct sum of finite point configurations 83 3 Combinatorial point configurations 91 3.1 Combinatorial configurations and their collections 92 3.2 Euclidean combinatorial configurations 95 3.3 Typology of combinatorial point configurations 99 3.4 Illustrative examples 103 3.5 Special cases of EMPCs and EPMPCs 106 4 Multipermutation point configurations 109 4.1 The entire multipermutation point configuration 109 4.2 The entire permutation point configuration 122 4.3 The EPPC En 126 4.4 The entire special multipermutation point configuration 131 4.5 The EBPC Bn (m) 137 4.6 The EMPC E′ n3(G) 142 4.7 Simple multipermutohedra 145 4.8 Combinatorically equivalent multipermutohedra 147 4.9 Illustration of Enk(G) and Πnk(G) (n = 3, 4) 148 5 Partial multipermutation point configurations 153 5.1 The entire partial multipermutation point configuration 153 5.2 The entire partial permutation point configuration 167 5.3 The entire unbounded partial permutation point configuration 170 5.4 The EPMPC En n+1,k(G) 173 5.5 The entire special partial multipermutation point configuration 179 5.6 The ESPPC Enn+1,2(G) 188 5.7 Vertex-located EPMPCs 190 5.8 The EBPPC Bn (m1,m2) 193 5.9 The EUBPPC Bn 200 5.10 Combinatorically equivalent partial multipermutohedra 204 5.11 Illustration of Enηk(G) and Πnηk(G) (n = 2, 3) 208 Conclusion 215 Bibliography 219