CornerMajorIndex

The sum, over the corners, of the step number along the boundary path from the bottom left: (number of parts - i) + the i-th part.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000770via IntegerPartition✓↗

CornerMajorIndex(partition)The sum of the positions of the corners, counted from the bottom row.

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.
• For (8, 6, 6, 4, 3, 3) the corners are at steps 3, 6, 9 and 13, so the value is 31 (FindStat's St000770 example).

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Sum(Map(Function(
                                           Length(_x) - i + _x[i] if i ==
                                                                     Length(_x) || _x[i] >
                                                                         _x[i +
                                                                            1] else 0,
                                           i,
                                         ), Range(1, Length(_x))))

Examples

Basic #

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

live ↗

In$\mathrm{CornerMajorIndex}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$10$

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

live ↗

In$\mathrm{CornerMajorIndex}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$5$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000770

FindStat's value for [3]

live ↗

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

Known value · FindStat St000770

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{CornerMajorIndex})$Out$4$

Known value · FindStat St000770

About

Examples	5
Crosswalk	FindStat