LeftTunnels

Tunnels (an up step with its matching down step) left of the middle: positions summing to less than the length plus one.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000120via DyckPath✓↗

LeftTunnels(path)Tunnels (an up step with its matching down step) left of the middle: positions summing to less than the length plus one.

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.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (m: list<integer>) => Length(Filter(
                                                                     Range(
                                                                            1,
                                                                            Length(_x),
                                                                          ),
                                                                     Function(_x[p] ==
                                                                              0 &&
                                                                              p +
                                                                              m[p] <
                                                                              Length(_x) +
                                                                              1, p),
                                                                   )),
                                Apply(
                                       (h: list<integer>) => Map(Function(
                                                                           Max(Filter(
                                                                                       Range(1, p - 1),
                                                                                       Function(_x[q] == 1 &&
                                                                                                h[q] ==
                                                                                                h[p - 1], q),
                                                                                     )) if _x[p] == 0 else 0,
                                                                           p,
                                                                         ), Range(1, Length(_x))),
                                       Map(
                                            Function(Sum(Map(
                                                              Function(
                                                                        2 *
                                                                        _x[k] - 1,
                                                                        k,
                                                                      ),
                                                              Range(1, i),
                                                            )), i),
                                            Range(1, Length(_x)),
                                          ),
                                     ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{LeftTunnels}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$2$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000120

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

live ↗

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

Known value · FindStat St000120

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

live ↗

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

Known value · FindStat St000120

About

Examples	5
Crosswalk	FindStat