Excedances

The number of excedances of $p$: positions $i$ with $p_i > i$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPermutation↗Permutation↗MathWorldPermutation↗WikidataQ161519↗Encyclopedia of MathematicsPermutation↗nLabpermutation↗Britannicatopic/permutation↗FindStatSt000155via Permutation✓↗St001864via SignedPermutation✓↗

Excedances(p)the excedance count of a one-line permutation $p$

Typed signatures

Type	Package	Applies
(permutation) -> integer	combinatorics	always

Domain: Permutation statistics

Details

• Equidistributed with Descents over $S_n$: both are Eulerian statistics
• A fixed point is neither an excedance nor an antiexcedance, so $\mathrm{Excedances} + \mathrm{Antiexcedances} + \mathrm{FixedPoints} = n$
• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] > i, i),
                                     ))

Examples

Basic #

only position 1: $3 > 1$

live ↗

In$\mathrm{Excedances}({\operatorname{Permutation}}([3, 1, 2]))$Out$1$

the identity has none

live ↗

In$\mathrm{Excedances}({\operatorname{Permutation}}([1, 2, 3]))$Out$0$

the one-line word $312$

live ↗

In$\mathrm{Excedances}({\operatorname{Permutation}}([3, 1, 2]))$Out$1$

the one-line word $2413$

live ↗

In$\mathrm{Excedances}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$2$

Properties #

excedances, antiexcedances and fixed points partition the $n$ positions

live ↗

In$\mathrm{Excedances}({\operatorname{Permutation}}([2, 3, 1]))+\mathrm{Antiexcedances}({\operatorname{Permutation}}([2, 3, 1]))+\mathrm{FixedPoints}({\operatorname{Permutation}}([2, 3, 1]))=3$Out$\mathrm{True}$

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{Excedances})$Out$0$

Known value · FindStat St000155

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{Excedances})$Out$0$

Known value · FindStat St000155

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{Excedances})$Out$1$

Known value · FindStat St000155

About

Examples	8
Crosswalk	Wikipedia ×2MathWorldWikidataEncyclopedia of MathematicsnLabBritannicaFindStat ×2

See also: Antiexcedances, Descents, SymmetricGroup