A permutation run is a maximal consecutive increasing block in a permutation, also called a run (Graham et al. 1994) or sometimes a rise (Comtet 1974, p. 241).
For a positive integer, the increasing permutation
of
elements has one run, whereas the decreasing permutation
has
runs, each of length 1. The number of permutations
of
with exactly
runs is denoted
(Comtet 1974, p. 241).
For example, the permutation contains the single run , while contains the two runs and . The following table lists the permutation runs for each
permutation of .
permutation runs
,
,
,
,
, ,
The number
of permutations of length with exactly runs is directly related to the number of permutation
ascents. A permutation with runs has ascents, and its reversal has ascents (Comtet 1974, p. 241; Skiena 1990, p. 31),
giving
In the limit as the length of a uniformly random permutation tends to infinity, the expected value of the
length of the first run is smaller than that of the second run (Gassner 1967; Skiena
1990, p. 30; Knuth 1998, pp. 40-41).
, the increasing permutation
of 
elements has one run, whereas the decreasing permutation
has 
runs, each of length 1. The number of permutations
of 
with exactly 
runs is denoted 
(Comtet 1974, p. 241).
contains the single run 
, while 
contains the two runs 
and 
. The following table lists the permutation runs for each
permutation 
of 
.
, 
, 
, 
, 
, 
, 
of permutations of length 
with exactly 
runs is directly related to the number of permutation
ascents. A permutation with 
runs has 
ascents, and its reversal has 
ascents (Comtet 1974, p. 241; Skiena 1990, p. 31),
giving
is an Eulerian number.
