Some relations on prefix reversal generators of the symmetric and hyperoctahedral group

Abstract

The symmetric group Sn and the group of signed permutations Bn (also referred to as the hyperoctahedral group) can be generated by prefixreversal permutations. A natural question is to determine the order of the “Coxeter-like” products formed by multiplying two generators, and in general, the relations satisfied by the prefix-reversal generators (also known as pancake generators or pancake flips). The order of these products is related to the length of certain cycles in the pancake and burnt pancake graphs. Using this connection, we derive a description of the order of the product of any two of these generators from a result due to Konstantinova and Medvedev. We provide a partial description of the order of the product of three generators when one of the generators is the transposition (1, 2). Furthermore, we describe the order of the product of two prefix-reversal generators in the hyperoctahedral group and give connections to the length of certain cycles in the burnt pancake graph.