  A permutation $\sigma$ describing the relative orders of the first $n$
iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is
called an \emph{order pattern}. For fixed $f$ and $n$, measuring the points
$x\in I$ (according to Lebesgue measure) that generate the order pattern
$\sigma$ gives a probability distribution $\mu_n(f)$ on the set of length $n$
permutations. We study the distributions that arise this way for various
classes of functions $f$.
  Our main results treat the class of measure preserving functions. We obtain
an exact description of the set of realizable distributions in this case: for
each $n$ this set is a union of open faces of the polytope of flows on a
certain digraph, and a simple combinatorial criterion determines which faces
are included. We also show that for general $f$, apart from an obvious
compatibility condition, there is no restriction on the sequence $\{\mu_n(f)\}$
for $n=1,2,...$.
  In addition, we give a necessary condition for $f$ to have \emph{finite
exclusion type}, i.e., for there to be finitely many order patterns that
generate all order patterns not realized by $f$. Using entropy we show that if
$f$ is piecewise continuous, piecewise monotone, and either ergodic or with
points of arbitrarily high period, then $f$ cannot have finite exclusion type.
This generalizes results of S. Elizalde.

Combinatorics;Dynamical Systems