LongTunnels

The longest sequences of consecutive tunnels: the semilength less the valleys.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000443via DyckPath✓↗St001007via DyckPath✓↗St001187via DyckPath✓↗St001224via DyckPath✓↗

LongTunnels(path)The longest sequences of consecutive tunnels: the semilength less the valleys.

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.
• Tunnels meet end to start exactly at valleys, and each valley joins one pair, so merging them leaves one long tunnel per tunnel not joined.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 2 else Length(_x) / 2 - Count(Filter(
                                                        Range(1, Length(_x) - 1),
                                                        Function(_x[i] == 0 && _x[i + 1] ==
                                                                               1, i),
                                                      ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000443

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

live ↗

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

Known value · FindStat St000443

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

live ↗

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

Known value · FindStat St000443

About

Examples	5
Crosswalk	FindStat ×4