AdjointInvariantDescents

The descents of the invariant in a tensor power of the adjoint representation of GL2 that the path indexes, read over its step pairs.

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AdjointInvariantDescents(path)The descents of the invariant in a tensor power of the adjoint representation of GL2 that the path indexes, read over its step pairs.

Typed signatures

Type	Package	Applies
(dyck_path) -> number	combinatorics	always

Domain: Dyck path statistics

Details

• Defined over DyckPath as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a DyckPath 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 St001104. The step pairs are up-up, up-down, down-up or down-down; a descent is a pair after which the next pair breaks the pattern, an up-up pair judged against the level reached.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 4 else Count(Filter(
                                       Range(1, Length(_x) / 2 - 1),
                                       Function(2 * _x[2p - 1] + _x[2p] == 3 &&
                                                (2 * _x[2 * (p + 1) - 1] + _x[2 *
                                                                              (p +
                                                                               1)] ==
                                                 1 || 2 * _x[2 * (p + 1) - 1] +
                                                     _x[2 * (p + 1)] == 0 && Count(Filter(
                                                                            Range(1, Length(_x) / 2),
                                                                            Function(q <= p &&
                                                                                     2 *
                                                                                     _x[2q - 1] +
                                                                                     _x[2q] ==
                                                                                     3, q),
                                                                          )) -
                                                              Count(Filter(
                                                                            Range(1, Length(_x) / 2),
                                                                            Function(q < p &&
                                                                                     2 *
                                                                                     _x[2q - 1] +
                                                                                     _x[2q] ==
                                                                                     0, q),
                                                                          )) > 1) || 2 * _x[2p - 1] + _x[2p] == 2 &&
                                                    (2 * _x[2 * (p + 1) - 1] +
                                                     _x[2 * (p + 1)] < 2 || 2 * _x[2 * (p + 1) -
                                                                    1] + _x[2 *
                                                                           (p +
                                                                            1)] > 2) || 2 * _x[2p - 1] + _x[2p] ==
                                                        1 && 2 * _x[2 * (p + 1) -
                                                                   1] +
                                                            _x[2 * (p + 1)] == 0, p),
                                     ))

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{AdjointInvariantDescents}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$0$

the step word $UDUUDD$

live ↗

In$\mathrm{AdjointInvariantDescents}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$1$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001104

FindStat's value for [1,0,1,0]

live ↗

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

Known value · FindStat St001104

FindStat's value for [1,1,0,0]

live ↗

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

Known value · FindStat St001104

About

Examples	5
Crosswalk	FindStat