MinimalRise

The length of the shortest rise.

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MinimalRise(path)The length of the shortest rise.

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) < 2 else Min(Apply(
                                    (u: list<integer>) => Map(
                                                               Function(
                                                                         u[1] if k == 1 else u[k] -
                                                                                             u[k - 1],
                                                                         k,
                                                                       ),
                                                               Range(
                                                                      1,
                                                                      Length(u),
                                                                    ),
                                                             ),
                                    Map(
                                         Function(Count(Filter(
                                                                Range(1, i),
                                                                Function(_x[k] ==
                                                                         1, k),
                                                              )), i),
                                         Filter(
                                                 Range(1, Length(_x) - 1),
                                                 Function(
                                                           _x[i] == 1 && _x[i +
                                                                            1] ==
                                                                         0,
                                                           i,
                                                         ),
                                               ),
                                       ),
                                  ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000655

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

live ↗

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

Known value · FindStat St000655

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

live ↗

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

Known value · FindStat St000655

About

Examples	5
Crosswalk	FindStat