DurfeeCube

The side of the largest cube in the array: the most diagonal entries d with the d-th at least d.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001451via PlanePartition✓↗

DurfeeCube(planePartition)The side of the largest cube in the array: the most diagonal entries d with the d-th at least d.

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 St001451, the plane partition analogue of the Durfee square. Entries down the diagonal never increase, so once one falls short of its place every later one does.

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 rowIds[u] == columnIds[u] &&
                                                                                                                                                                                                        reading[u] >= rowIds[u] 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$\mathrm{DurfeeCube}({\operatorname{PlanePartition}}([[4, 3, 1], [3, 1], [1]]))$Out$1$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St001451

FindStat's value for [[2]]

live ↗

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

Known value · FindStat St001451

FindStat's value for [[3]]

live ↗

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

Known value · FindStat St001451

About

Examples	4
Crosswalk	FindStat