RimHooksRemovedToTwoCore

The 2-rim hooks removed to reach the 2-core: the size less the core's, over 2, the core read off the abacus.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000698via IntegerPartition✓↗

RimHooksRemovedToTwoCore(partition)The number of 2-rim hooks removed to reach the 2-core: the size less the core's, over 2. The core's size is read off the abacus, from how many beads sit on each runner.

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 (Sum(_x) - (Sum(Map(
                                              Function(Count(Filter(
                                                                     Range(
                                                                            1,
                                                                            Length(_x),
                                                                          ),
                                                                     Function((_x[i] +
                                                                               (Length(_x) - i)) %
                                                                              2 ==
                                                                              r, i),
                                                                   )) *
                                                       (r + 2 * (Count(Filter(
                                                                               Range(1, Length(_x)),
                                                                               Function((_x[i] +
                                                                                         (Length(_x) - i)) %
                                                                                        2 ==
                                                                                        r, i),
                                                                             )) -
                                                                 1) / 2), r),
                                              Range(0, 1),
                                            )) - Length(_x) * (Length(_x) - 1) /
                                                2)) / 2

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000698

FindStat's value for [3]

live ↗

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

Known value · FindStat St000698

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{RimHooksRemovedToTwoCore})$Out$2$

Known value · FindStat St000698

About

Examples	5
Crosswalk	FindStat