StaircaseSizeOfCode

The largest k with a subsequence of the Lehmer code whose j-th entry from the right is at least j.

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StaircaseSizeOfCode(p)The largest k with a subsequence of the Lehmer code whose j-th entry from the right is at least j.

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.
• Scanning the code from the right and taking each entry that exceeds the count so far is optimal.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Apply(
                                (c: list<integer>) => Fold(
                                                            Function(
                                                                      a + 1 if c[b] >
                                                                               a else a,
                                                                      a,
                                                                      b,
                                                                    ),
                                                            0,
                                                            Range(
                                                                   Length(_x),
                                                                   1,
                                                                   -1,
                                                                 ),
                                                          ),
                                Map(
                                     Function(Count(Filter(
                                                            Range(
                                                                   i + 1,
                                                                   Length(_x),
                                                                   1,
                                                                 ),
                                                            Function(_x[j] <
                                                                     _x[i], j),
                                                          )), i),
                                     Range(1, Length(_x)),
                                   ),
                              )

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000662

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000662

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000662

About

Examples	6
Crosswalk	FindStat