DoubleStepVisiblePoints

Points below the path, at the parity of the path's height there, from which the north-east diagonal meets a double fall and the north-west diagonal meets a double rise.

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DoubleStepVisiblePoints(path)Points below the path, at the parity of the path's height there, from which the north-east diagonal meets a double fall and the north-west diagonal meets a double 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.
• A point (v, j) has j = h - 2m under the path's vertex v of height h. Its north-east diagonal meets the path at a double fall (v, h') with v - j = v' - h'; its north-west diagonal at a double rise with v + j = v' + h'.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 2 else Apply(
                                (vp: list<integer>) => Apply(
                                                              (doubleRises: list<integer>) => Apply(
                                                                                                     (doubleFalls: list<integer>) => Sum(Map(Function(Apply(
                                                                                                                                                             (h: number) => 0 if h < 2 else Count(Filter(
                                                                                                                                                                                                          Range(1, h),
                                                                                                                                                                                                          Function(2m <= h &&
                                                                                                                                                                                                                   Count(Filter(doubleRises, Function(s ==
                                                                                                                                                                                                                                                      v +
                                                                                                                                                                                                                                                      (h - 2m), s))) >
                                                                                                                                                                                                                   0 &&
                                                                                                                                                                                                                   Count(Filter(doubleFalls, Function(s ==
                                                                                                                                                                                                                                                      v -
                                                                                                                                                                                                                                                      (h - 2m), s))) >
                                                                                                                                                                                                                   0, m),
                                                                                                                                                                                                        )),
                                                                                                                                                             vp[v + 1],
                                                                                                                                                           ), v), Range(0, Length(_x)))),
                                                                                                     Map(
                                                                                                          Function(i -
                                                                                                                   Sum(Map(Function(2 * _x[k] -
                                                                                                                                    1, k), Range(1, i))), i),
                                                                                                          Filter(
                                                                                                                  Range(1, Length(_x) - 1),
                                                                                                                  Function(_x[i] == 0 &&
                                                                                                                           _x[i + 1] ==
                                                                                                                           0, i),
                                                                                                                ),
                                                                                                        ),
                                                                                                   ),
                                                              Map(
                                                                   Function(i +
                                                                            Sum(Map(
                                                                                     Function(2 * _x[k] -
                                                                                              1, k),
                                                                                     Range(1, i),
                                                                                   )), i),
                                                                   Filter(
                                                                           Range(1, Length(_x) - 1),
                                                                           Function(_x[i] == 1 &&
                                                                                    _x[i + 1] ==
                                                                                    1, i),
                                                                         ),
                                                                 ),
                                                            ),
                                Join(
                                      [0],
                                      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$\mathrm{DoubleStepVisiblePoints}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000683

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

live ↗

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

Known value · FindStat St000683

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

live ↗

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

Known value · FindStat St000683

About

Examples	5
Crosswalk	FindStat