HookFactorizationPrefixLength

The length of the leading increasing factor p of the hook factorization p t1 ... tk.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000338via Permutation✓↗

HookFactorizationPrefixLength(p)The length of the leading increasing factor p of the hook factorization p t1 ... tk.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• Hook factors are cut from the right, each from a descent top.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

Length(_x) if Length(_x) < 2 else Fold(
                                        Function(
                                                  b if b <= a - 2 && _x[b] >
                                                                     _x[b + 1] else a,
                                                  a,
                                                  b,
                                                ),
                                        Length(_x) + 1,
                                        Range(Length(_x) - 1, 1, -1),
                                      ) - 1

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{HookFactorizationPrefixLength}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $312$, as a plain list

live ↗

In$\mathrm{HookFactorizationPrefixLength}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000338

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000338

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{HookFactorizationPrefixLength})$Out$0$

Known value · FindStat St000338

About

Examples	6
Crosswalk	FindStat