UlamDistance

The entries to remove to leave an increasing sequence: n less the length of a longest increasing subsequence.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000829via Permutation✓↗

UlamDistance(p)The entries to remove to leave an increasing sequence: n less the length of a longest increasing subsequence.

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.
• FindStat takes the longest increasing subsequence of the inverse, which has the same length.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Length(_x) - Count(Fold(Function(Apply(
                                                                 Function(
                                                                           Join(a, [b]) if k > Count(a) else ReplaceAt(a, k, b),
                                                                           k,
                                                                         ),
                                                                 Count(Filter(
                                                                               a,
                                                                               Function(t < b, t),
                                                                             )) +
                                                                 1,
                                                               ), a, b), [], _x))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000829

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000829

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St000829

About

Examples	6
Crosswalk	FindStat