Antiexcedances

The number of antiexcedances of $p$: positions $i$ with $p_i < i$.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000703via Permutation✓↗

Antiexcedances(p)the antiexcedance count of a one-line permutation $p$

Typed signatures

Type	Package	Applies
(permutation) -> integer	combinatorics	always

Domain: Permutation statistics

Details

• Equidistributed with Excedances over $S_n$: inverting a permutation swaps its excedance and antiexcedance counts
• Never counts a fixed point, same as Excedances
• 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 #

positions 2 and 3: $1 < 2$ and $2 < 3$

live ↗

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

the identity has none

live ↗

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

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

positions 3 and 4: $2 < 3$ and $1 < 4$

live ↗

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

FindStat's value for [1]

live ↗

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

Known value · FindStat St000703

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000703

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000703

About

Examples	8
Crosswalk	FindStat

See also: Excedances, Descents, SymmetricGroup