StasinskiVollLength

Half the pairs -n <= i < j <= n of opposite parity with p(i) > p(j), where p(-k) = -p(k) and p(0) = 0.

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StasinskiVollLength(p)Half the pairs -n <= i < j <= n of opposite parity with p(i) > p(j), where p(-k) = -p(k) and p(0) = 0.

Typed signatures

Type	Package	Applies
(signed_permutation) -> number	combinatorics	always

Domain: Signed permutation statistics

Details

• Defined over SignedPermutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SignedPermutation 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.
• FindStat's St001823. The map (i, j) -> (-j, -i) pairs the counted pairs off, so counting those with i + j > 0, i = j - 1, j - 3, ..., 1 - j, gives the half.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Max(Sum(Map(Function(Count(Filter(
                                                            Range(0, j - 1),
                                                            Function((_x[j - 1 -
                                                                         2t] if j -
                                                                              1 -
                                                                              2t >
                                                                              0 else 0 if j - 1 -
                                                                                         2t ==
                                                                                         0 else -_x[-(j - 1 -
                                                                                                     2t)]) >
                                                                     _x[j], t),
                                                          )), j), Range(
                                                                        1,
                                                                        Length(_x),
                                                                      ))), 0)

Examples

Basic #

the signed word $2\,\bar{1}\,\bar{3}$

live ↗

In$\mathrm{StasinskiVollLength}({\operatorname{SignedPermutation}}([2, -1, -3]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001823

FindStat's value for [-1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([-1]), \text{StasinskiVollLength})$Out$1$

Known value · FindStat St001823

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001823

About

Examples	4
Crosswalk	FindStat