Cycles of lentgth 9 in the Pancake graph

A cycle $C_l$ of length $l$, where $6\le l\le n!$, can be embedded in the Pancake graph $P_n$, $n\ge3$, that is the Cayley graph on the symmetric group with the generating set of all prefix-reversals. The algebraic characterization of cycles of length six and seven via products of generating elements is known. We continue to study odd cycles. The explicit description of cycles of length nine by means of 10 canonical forms is given. It is also proved that each of vertices of $P_n$, $n\ge4,$ belongs to $\frac{8n^3-45n^2+61n-12}2$ cycles of length nine. In general, there are $O(n!\,n^3)$ different cycles of length nine in the graph. Ill. 5, tab. 1, bibliogr. 10.