BoxesBelowAtLeastTwoPeaks

The boxes between the path and the diagonal that lie right of and below at least two peaks.

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BoxesBelowAtLeastTwoPeaks(path)The boxes between the path and the diagonal that lie right of and below at least two peaks.

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 Apply(
                                (pu: list<integer>) => Apply(
                                                              (pd: list<integer>) => Apply(
                                                                                            (downs: list<integer>) => Sum(Map(
                                                                                                                               Function(0 if downs[k] - k <
                                                                                                                                             k + 1 else Count(Filter(
                                                                                                                                                                 Range(k, downs[k] - k -
                                                                                                                                                                          1),
                                                                                                                                                                 Function(Count(Filter(Range(1, Length(pu)), Function(pd[q] < k &&
                                                                                                                                                                                                                      pu[q] > c, q))) >=
                                                                                                                                                                          2, c),
                                                                                                                                                               )), k),
                                                                                                                               Range(1, Length(_x) / 2),
                                                                                                                             )),
                                                                                            Filter(
                                                                                                    Range(1, Length(_x)),
                                                                                                    Function(_x[i] == 0, i),
                                                                                                  ),
                                                                                          ),
                                                              Map(
                                                                   Function(i -
                                                                            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),
                                                                         ),
                                                                 ),
                                                            ),
                                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$\mathrm{BoxesBelowAtLeastTwoPeaks}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$0$

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000980

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

live ↗

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

Known value · FindStat St000980

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

live ↗

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

Known value · FindStat St000980

About

Examples	5
Crosswalk	FindStat