Editing Permutation (section)

===Ascents, descents, runs, exceedances, records===
{{anchor|Descents}}
An ''ascent'' of a permutation&nbsp;''σ'' of ''n'' is any position ''i''&nbsp;&lt;&nbsp;''n'' where the following value is bigger than the current one. That is, ''i'' is an ascent if <math>\sigma(i)<\sigma(i{+}1)</math>. For example, the permutation 3452167 has ascents (at positions) 1, 2, 5, and 6.

Similarly, a ''descent'' is a position ''i''&nbsp;&lt;&nbsp;''n'' with <math>\sigma(i)>\sigma(i{+}1)</math>, so every ''i'' with <math>1 \leq i<n</math> is either an ascent or a descent.

An ''ascending run'' of a permutation is a nonempty increasing contiguous subsequence that cannot be extended at either end; it corresponds to a maximal sequence of successive ascents (the latter may be empty: between two successive descents there is still an ascending run of length&nbsp;1). By contrast an ''increasing subsequence'' of a permutation is not necessarily contiguous: it is an increasing sequence obtained by omitting some of the values of the one-line notation.
For example, the permutation 2453167 has the ascending runs 245, 3, and 167, while it has an increasing subsequence 2367.

If a permutation has ''k''&nbsp;−&nbsp;1 descents, then it must be the union of ''k'' ascending runs.{{sfn|Bóna|2004|p=4f}}

The number of permutations of ''n'' with ''k'' ascents is (by definition) the [[Eulerian number]] <math>\textstyle\left\langle{n\atop k}\right\rangle</math>; this is also the number of permutations of ''n'' with ''k'' descents. Some authors however define the Eulerian number <math>\textstyle\left\langle{n\atop k}\right\rangle</math> as the number of permutations with ''k'' ascending runs, which corresponds to {{math|''k'' − 1}} descents.{{sfn|Bona|2012|pages=4–5}}

An exceedance of a permutation ''σ''<sub>1</sub>''σ''<sub>2</sub>...''σ''<sub>''n''</sub> is an index ''j'' such that {{math|''σ''<sub>''j''</sub> > ''j''}}. If the inequality is not strict (that is, {{math|''σ''<sub>''j''</sub> ≥ ''j''}}), then ''j'' is called a ''weak exceedance''. The number of ''n''-permutations with ''k'' exceedances coincides with the number of ''n''-permutations with ''k'' descents.{{sfn|Bona|2012|page=25}}

A ''record'' or ''left-to-right maximum'' of a permutation ''σ'' is an element ''i'' such that ''σ''(''j'') < ''σ''(''i'') for all ''j < i''.