DecompositionNumber

The k with {p(1), ..., p(k)} = {1, ..., k}: the blocks of p as a direct sum.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000056via Permutation✓↗

DecompositionNumber(p)The k with {p(1), ..., p(k)} = {1, ..., k}: the blocks of p as a direct sum.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation 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.
• One more than GlobalAscents; also called the block number.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(Max(Map(
                                                         Function(_x[j], j),
                                                         Range(1, i),
                                                       )) == i, i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000056

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000056

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000056

About

Examples	5
Crosswalk	FindStat