DiagonalArmSum

The sum of the arms of the cells on the main diagonal.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000319via IntegerPartition✓↗St000320via IntegerPartition✓↗

DiagonalArmSum(partition)The sum of the arms of the cells on the main diagonal.

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.
• FindStat's spin (St000319) counts the crossings of the border strips and its dinv adjustment (St000320) sums over their starting columns; each strip contributes its first part less 1, which is the arm of one diagonal cell, and the two agree on every row.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

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

Examples

Basic #

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

live ↗

In$\mathrm{DiagonalArmSum}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$3$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000319

FindStat's value for [2]

live ↗

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

Known value · FindStat St000319

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{DiagonalArmSum})$Out$2$

Known value · FindStat St000319

About

Examples	5
Crosswalk	FindStat ×2