SmallestCylindricShift

The least shift s at which each entry of the last row is smaller than the first row's entry s columns on.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001993via StandardTableau✓↗

SmallestCylindricShift(tableau)The least shift s at which each entry of the last row is smaller than the first row's entry s columns on.

Typed signatures

Type	Package	Applies
(standard_tableau) -> number	combinatorics	always

Domain: Tableau statistics

Details

• Defined over StandardTableau as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a StandardTableau 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.
• Shifts below width - last row length cannot work, and a shift that works keeps working one column on, so the least one is that bound plus the shifts above it that fail (FindStat's St001993). The empty tableau gives 0.

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(_x) < 1 else Apply((reading: list<integer>) => Apply(
                                                                  (lengths: list<integer>) => Apply((width: integer) => Apply((lastLength: integer) => width - lastLength +
                                                                                                                                                       Max(Sum(Map(Function(1 if Max(Sum(Map(Function(1 if reading[Length(reading) - lastLength +
                                                                                                                                                                                                                   j] >=
                                                                                                                                                                                                           reading[i + j] else 0, j), Range(1, Min(lastLength, width - i), 1))), 0) >
                                                                                                                                                                                 0 else 0, i), Range(width - lastLength, width - 1, 1))), 0), lengths[Length(_x)]), lengths[1]),
                                                                  Map(
                                                                       Function(Length(r), r),
                                                                       _x,
                                                                     ),
                                                                ), Flatten(_x))

Examples

Basic #

the standard tableau $[[1,2,4],[3,6],[5]]$

live ↗

In$\mathrm{SmallestCylindricShift}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$3$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St001993

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

live ↗

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

Known value · FindStat St001993

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

live ↗

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

Known value · FindStat St001993

About

Examples	4
Crosswalk	FindStat