FirstColumnInitialSequenceLength

The largest k such that the first column starts 1, 2, ..., k.

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FirstColumnInitialSequenceLength(tableau)The largest k such that the first column starts 1, 2, ..., k.

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.
• FindStat's title is the index of the last row whose first entry is the row number (St000745).

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(_x) < 1 else Apply(
                                (firsts: list<integer>) => Max(Sum(Map(
                                                                        Function(
                                                                                  1 if firsts[i] == i else 0,
                                                                                  i,
                                                                                ),
                                                                        Range(1, Length(_x)),
                                                                      )), 0),
                                Map(Function(r[1], r), _x),
                              )

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000745

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

live ↗

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

Known value · FindStat St000745

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

live ↗

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

Known value · FindStat St000745

About

Examples	4
Crosswalk	FindStat