Abstract An odd wheel graph is a formed by connecting new vertex to all vertices of an cycle. We answer question Rosenfeld and Le showing that wheels cannot be drawn in the plane so lengths edges are integers.

We define and study odd analogues of classical geometric combinatorial objects associated to permutations, namely Schubert varieties, diagrams, inversion sets. show that there is a bijection between sets permutations acyclic orientations the Turán graph, dimension variety permutation length permutation, give several necessary conditions for subset [n]×[n] be diagram permutation. also sign-twist...