FrobeniusRank

The fewest border strips that the shape divides into.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001597via SkewPartition✓↗

FrobeniusRank(shape)The fewest border strips that the shape divides into.

Typed signatures

Type	Package	Applies
(skew_partition) -> number	combinatorics	always

Domain: Skew partition statistics

Details

• Defined over SkewPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SkewPartition 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 St001597. A strip is a path of cells going up or right, so the fewest strips are the cells less the most pairs of neighbors that no cell takes twice as the one before and as the one after. Those pairs lie between consecutive diagonals, in runs, and a run of m cells gives floor(m / 2). A connected shape of one strip is a border strip. The empty shape gives 0.

Definition

Epsil, as the implementation runs it.

on SkewPartition, over _x

0 if Length(_x[1]) == 0 else Apply((lam: list<integer>) => Apply((raw: list<integer>) => Apply(
                                                                                                (mu: list<integer>) => Max(Sum(lam), 0) -
                                                                                                                       Max(Sum(mu), 0) -
                                                                                                                       Max(Sum(Map(Function(Fold(
                                                                                                                                                  Function([
                                                                                                                                                             a[1] + 1 if b == 1 else 0,
                                                                                                                                                             a[2] + 1 if b == 1 &&
                                                                                                                                                                         a[1] % 2 ==
                                                                                                                                                                         1 else a[2],
                                                                                                                                                           ], a, b),
                                                                                                                                                  [0, 0],
                                                                                                                                                  Flatten(Map(Function([1 if mu[r] <=
                                                                                                                                                                             r - 1 +
                                                                                                                                                                             k &&
                                                                                                                                                                             r - 1 +
                                                                                                                                                                             k <
                                                                                                                                                                             lam[r] else 0, 1 if mu[r] <=
                                                                                                                                                                                           r + k &&
                                                                                                                                                                                           r + k <
                                                                                                                                                                                           lam[r] else 0], r), Range(1, Length(lam)))),
                                                                                                                                                )[2], k), Range(0 - Length(lam), Max(lam)))), 0),
                                                                                                Map(
                                                                                                     Function(raw[i] if i <= Length(raw) else 0, i),
                                                                                                     Range(1, Length(lam)),
                                                                                                   ),
                                                                                              ), _x[2]), _x[1])

Examples

Basic #

the skew partition $[[3,1],[1]]$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001597

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

live ↗

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

Known value · FindStat St001597

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SkewPartition}}([[3], []]), \text{FrobeniusRank})$Out$1$

Known value · FindStat St001597

About

Examples	4
Crosswalk	FindStat