RightDescents

Indices 0 <= k < n with s(k) > s(k+1), where s is the inverse of p and s(0) = 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001896via SignedPermutation✓↗

RightDescents(p)Indices 0 <= k < n with s(k) > s(k+1), where s is the inverse of p and s(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 St001896: the left descents of the inverse. The inverse sends k to the position of k or -k, signed as the entry is.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] == -1, i),
                                     )) +
                         (0 if Length(_x) < 2 else Count(Filter(
                                                                 Range(
                                                                        1,
                                                                        Length(_x) -
                                                                        1,
                                                                      ),
                                                                 Function((1 - 2 *
                                                                               Count(Filter(
                                                                                             Range(1, Length(_x)),
                                                                                             Function(_x[i] == -d, i),
                                                                                           ))) *
                                                                          Max(Sum(Map(
                                                                                       Function(i if Abs(_x[i]) == d else 0, i),
                                                                                       Range(1, Length(_x)),
                                                                                     )), 0) > (1 -
                                                                              2 *
                                                                              Count(Filter(
                                                                                            Range(1, Length(_x)),
                                                                                            Function(_x[i] ==
                                                                                                     -(d + 1), i),
                                                                                          ))) *
                                                                             Max(Sum(Map(
                                                                                          Function(i if Abs(_x[i]) ==
                                                                                                        d + 1 else 0, i),
                                                                                          Range(1, Length(_x)),
                                                                                        )), 0), d),
                                                               )))

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001896

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001896

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001896

About

Examples	4
Crosswalk	FindStat