HookLengthOfLastDiagonalCell

The hook length of the last cell on the main diagonal: that of cell (d,d) for Durfee side d.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000460via IntegerPartition✓↗

HookLengthOfLastDiagonalCell(partition)The hook length of the last cell on the main diagonal: that of cell (d,d) for Durfee side d.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition 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 IntegerPartition, over _x

0 if Length(_x) < 1 else Apply(
                                (c: list<integer>) => Apply(
                                                             (d: number) => _x[d] -
                                                                            d +
                                                                            (c[d] -
                                                                             (d -
                                                                              1)),
                                                             Count(Filter(
                                                                           Range(1, Length(_x)),
                                                                           Function(_x[i] >= i, i),
                                                                         )),
                                                           ),
                                Map(
                                     Function(
                                               Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       _x[j] >=
                                                                       i,
                                                                       j,
                                                                     ),
                                                           )),
                                               i,
                                             ),
                                     Range(1, Max(_x)),
                                   ),
                              )

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{HookLengthOfLastDiagonalCell}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{HookLengthOfLastDiagonalCell}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$2$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000460

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{HookLengthOfLastDiagonalCell})$Out$2$

Known value · FindStat St000460

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{HookLengthOfLastDiagonalCell})$Out$3$

Known value · FindStat St000460

About

Examples	5
Crosswalk	FindStat