Inversions, cuts, and orientations (Q2640623)

A reorientation of an ordered set is an orientation of its covering graph. A pushdown is a reorientation Q of an ordered set P which can be obtained in the following way: Fix a maximal element a of P and reverse the edges of the diagram of P which have a as endpoint. A sequence of pushdowns is called an inversion. The following theorem is proved: An ordered set is an inversion of the finite ordered set P iff the reversed edges can be partitioned into cuts of P. Here a subset E of edges of P (say E consists of the covering pairs \(a_ 1\succ b_ 1\), \(a_ 2\succ b_ 2\),...) is said to be a cut if there is no element a in P which is connected to some \(a_ i\) and to some \(b_ j\) in the diagram obtained from P by removing all of the edges of E. Applications to the enumeration and complexity of reorientations are given.