A grid poset -or grid for short -is a product of chains. We ask, what does a random linear extension of a grid look like? In particular, we show that the average "jump number," i.e., the number of times that two consecutive elements in a linear extension are incomparable in the poset, is close to its maximum possible value. The techniques employed rely on entropy arguments. We finish with several interesting questions about this wide-open area.
There is some recent interest in determining the average jump number of a grid REF , which is equivalent to determining the success probability of the grid.
Random Linear Extensions of Grids
arXiv: Combinatorics
Mathematics