RemovableHorizontalDominoes

Parts from which 2 can be subtracted and leave a partition: rows ending in a removable horizontal domino.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000256via IntegerPartition✓↗

RemovableHorizontalDominoes(partition)Parts from which 2 can be subtracted and leave a partition: rows ending in a removable horizontal domino.

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 Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] >= 2 if i == Length(_x) else _x[i] - 2 >=
                                                                                   _x[i + 1], i),
                                     ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000256

FindStat's value for [2]

live ↗

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

Known value · FindStat St000256

FindStat's value for [3]

live ↗

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

Known value · FindStat St000256

About

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