A permutation ascent is an index 
with 
for which 
in a permutation 
. For example, 
has three ascents, at indices
1, 2, and 3.
For a positive integer 
, the number of permutations
of length 
having 
ascents is given by the Eulerian number 
.
The total number of ascents in all permutations of 
elements is
giving the first few terms for 
, 2, ... as 0, 1, 6, 36, 240, 1800, 15120, ... (OEIS A001286).
A permutation of 
elements with 
ascents has 
increasing permutation runs.
Reversing a permutation with 
ascents gives one with 
ascents, so the number of permutations
with 
ascents also equals the number 
with 
permutation runs (Skiena
1990, p. 31), giving
See also
Eulerian Number,
Permutation,
Permutation Run
Explore with Wolfram|Alpha
References
Bona, M. Combinatorics of Permutations. Boca Raton, FL: Chapman & Hall/CRC, 2004.Graham,
R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley,
1994.Knuth, D. E. The
Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed.
Reading, MA: Addison-Wesley, 1998.Mannila, H. "Measures of Presortedness
and Optimal Sorting Algorithms." IEEE Trans. Comput. 34, 318-325,
1985. https://doi.org/10.1109/TC.1985.5009382.Skiena,
S. "Runs and Eulerian Numbers." §1.3.4 in Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, pp. 30-31, 1990.Sloane, N. J. A.
Sequence A001286/M4225 in "The On-Line
Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Permutation Ascent
Cite this as:
Weisstein, Eric W. "Permutation Ascent."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PermutationAscent.html
Subject classifications
