SelfEvacuatingTableauCount

The standard Young tableaux of the shape that evacuation fixes, equivalently its standard domino tableaux.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000618via IntegerPartition✓↗

SelfEvacuatingTableauCount(partition)The standard Young tableaux of the shape that evacuation fixes, equivalently its standard domino tableaux.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition 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.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Apply(
                                (t: list<integer>) => (1 if (Sum(_x) - Sum(_x) %
                                                                       2) / 2 ==
                                                            0 else Product(Range(
                                                                                 1,
                                                                                 (Sum(_x) -
                                                                                  Sum(_x) % 2) /
                                                                                 2,
                                                                               ))) *
                                                      Product(Map(
                                                                   Function(Product(Map(
                                                                                         Function((t[j] - t[i]) /
                                                                                                  2 if j < i &&
                                                                                                      (t[j] - t[i]) %
                                                                                                      2 ==
                                                                                                      0 else 1, j),
                                                                                         Range(1, Length(t)),
                                                                                       )), i),
                                                                   Range(
                                                                          1,
                                                                          Length(t),
                                                                        ),
                                                                 )) /
                                                      Product(Map(Function(
                                                                            1 if (j -
                                                                                  j % 2) /
                                                                                 2 ==
                                                                                 0 else Product(Range(1, (j -
                                                                                                         j % 2) /
                                                                                                        2)),
                                                                            j,
                                                                          ), t)) if Sum(Map(Function((j -
                                                                            j %
                                                                            2) /
                                                                           2, j), t)) -
                                                          (Count(Filter(
                                                                         t,
                                                                         Function(j % 2 ==
                                                                                  0, j),
                                                                       )) * (Count(Filter(t, Function(j % 2 ==
                                                                                                     0, j))) -
                                                                            1) /
                                                           2 + Count(Filter(
                                                                            t,
                                                                            Function(j % 2 ==
                                                                                     1, j),
                                                                          )) * (Count(Filter(t, Function(j % 2 ==
                                                                                                        1, j))) -
                                                                               1) /
                                                              2) == (Sum(_x) - Sum(_x) %
                                                                         2) / 2 else 0,
                                Map(
                                     Function(_x[i] + (Length(_x) - i), i),
                                     Range(1, Length(_x)),
                                   ),
                              )

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{SelfEvacuatingTableauCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{SelfEvacuatingTableauCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$3$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000618

FindStat's value for [2]

live ↗

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

Known value · FindStat St000618

FindStat's value for [3]

live ↗

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

Known value · FindStat St000618

About

Examples	5
Crosswalk	FindStat