RemovableBoxCount

The number of entries larger than both the entry to their right and the entry below.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001445via PlanePartition✓↗

RemovableBoxCount(planePartition)The number of entries larger than both the entry to their right and the entry below.

Typed signatures

Type	Package	Applies
(plane_partition) -> number	combinatorics	always

Domain: Plane partition statistics

Details

• Defined over PlanePartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a PlanePartition 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.
• FindStat's St001445. Taking one box off such an entry leaves a plane partition, and off any other does not.

Definition

Epsil, as the implementation runs it.

on PlanePartition, over _x

0 if Length(Flatten(_x)) < 1 else Apply((reading: list<integer>) => Apply(
                                                                           (lengths: list<integer>) => Apply(
                                                                                                              (rowIds: list<integer>) => Apply(
                                                                                                                                                (columnIds: list<integer>) => Max(Sum(Map(Function(1 if reading[u] >
                                                                                                                                                                                                        (reading[u + 1] if columnIds[u] < lengths[rowIds[u]] else 0) &&
                                                                                                                                                                                                        reading[u] >
                                                                                                                                                                                                        ((reading[u + lengths[rowIds[u]]] if columnIds[u] <=
                                                                                                                                                                                                                                             lengths[rowIds[u] + 1] else 0) if rowIds[u] < Length(_x) else 0) else 0, u), Range(1, Length(reading), 1))), 0),
                                                                                                                                                Flatten(Map(Function(Range(1, lengths[i], 1), i), Range(1, Length(_x)))),
                                                                                                                                              ),
                                                                                                              Flatten(Map(
                                                                                                                           Function(Map(Function(i, j), Range(1, lengths[i], 1)), i),
                                                                                                                           Range(1, Length(_x)),
                                                                                                                         )),
                                                                                                            ),
                                                                           Map(
                                                                                Function(Length(r), r),
                                                                                _x,
                                                                              ),
                                                                         ), Flatten(_x))

Examples

Basic #

the plane partition $[[4,3,1],[3,1],[1]]$

live ↗

In$\mathrm{RemovableBoxCount}({\operatorname{PlanePartition}}([[4, 3, 1], [3, 1], [1]]))$Out$6$

Properties #

FindStat's value for [[1]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PlanePartition}}([[1]]), \text{RemovableBoxCount})$Out$1$

Known value · FindStat St001445

FindStat's value for [[2]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PlanePartition}}([[2]]), \text{RemovableBoxCount})$Out$1$

Known value · FindStat St001445

FindStat's value for [[3]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PlanePartition}}([[3]]), \text{RemovableBoxCount})$Out$1$

Known value · FindStat St001445

About

Examples	4
Crosswalk	FindStat