On the permutation polytopes of some cyclic groups

Abstract In this article, we study the combinatorics of permutation polytopes of some cyclic groups, denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>P</m:mi> <m:mrow> <m:mo>⟨</m:mo> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mo>⟩</m:mo> </m:mrow> </m:math> P\langle g\rangle where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> g\in {S}_{n} . We give a formula on the dimension and the number of vertices of the smallest faces containing two given vertices. Then, we study the number of the smallest faces containing one fixed vertex and some other vertex which are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -dimensional. As a consequence, the diameter of the vertex-edge graph of some special permutation polytopes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>P</m:mi> <m:mrow> <m:mo>⟨</m:mo> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mo>⟩</m:mo> </m:mrow> </m:math> P\langle g\rangle is obtained. We also consider the reduced Gröbner basis of the toric ideal of some special <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>P</m:mi> <m:mrow> <m:mo>⟨</m:mo> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mo>⟩</m:mo> </m:mrow> </m:math> P\langle g\rangle and show that in this case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>P</m:mi> <m:mrow> <m:mo>⟨</m:mo> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mo>⟩</m:mo> </m:mrow> </m:math> P\langle g\rangle is compressed.