RecursiveStartMinimumCount

FindStat's aix: the number of times, peeling the word, the minimum of what is left is its first entry.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000873via Permutation✓↗

RecursiveStartMinimumCount(p)FindStat's aix: the number of times, peeling the word, the minimum of what is left is its first entry.

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.
• Recursively: the minimum first counts one and the rest is the word without it; the minimum last ends the walk; otherwise the walk goes on with the entries before the minimum (St000873).

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Fold(Function(a if a[1] > a[2] else Apply(
                                                                    (bottom: integer) => Apply(
                                                                                                (low: integer) => [a[1] + 1, a[2], a[3] + 1] if low == a[1] else [1, 0, a[3]] if low == a[2] else [a[1], low - 1, a[3]],
                                                                                                Max(Sum(Map(
                                                                                                             Function(j if _x[j] == bottom else 0, j),
                                                                                                             Range(a[1], a[2], 1),
                                                                                                           )), 0),
                                                                                              ),
                                                                    Min(Map(
                                                                             Function(_x[j], j),
                                                                             Range(a[1], a[2], 1),
                                                                           )),
                                                                  ), a, b), [
                                                                             1,
                                                                             Length(_x),
                                                                             0,
                                                                           ], Range(1, Length(_x)))[3]

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000873

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000873

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

live ↗

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

Known value · FindStat St000873

About

Examples	5
Crosswalk	FindStat