ShiftedStandardTableauCount

Standard tableaux of the shifted shape: the multinomial of the parts times, over pairs of parts, the difference over the sum. 0 when two parts are equal.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001593via IntegerPartition✓↗

ShiftedStandardTableauCount(partition)The number of standard tableaux of the shifted shape: the multinomial of the parts times the product over pairs of parts of (difference over sum). 0 when two parts are equal.

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

1 if Length(_x) < 1 else (1 if Sum(_x) == 0 else Product(Range(1, Sum(_x)))) *
                         Product(Map(
                                      Function(Product(Map(
                                                            Function(
                                                                      _x[j] -
                                                                      _x[i] if j < i else 1,
                                                                      j,
                                                                    ),
                                                            Range(1, Length(_x)),
                                                          )), i),
                                      Range(1, Length(_x)),
                                    )) / (Product(Map(Function(
                                                   1 if _x[i] == 0 else Product(Range(1, _x[i])),
                                                   i,
                                                 ), Range(1, Length(_x)))) *
                             Product(Map(
                                          Function(Product(Map(
                                                                Function(
                                                                          _x[j] +
                                                                          _x[i] if j <
                                                                              i else 1,
                                                                          j,
                                                                        ),
                                                                Range(
                                                                       1,
                                                                       Length(_x),
                                                                     ),
                                                              )), i),
                                          Range(1, Length(_x)),
                                        )))

Examples

Basic #

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

live ↗

In$\mathrm{ShiftedStandardTableauCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$7$

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

live ↗

In$\mathrm{ShiftedStandardTableauCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001593

FindStat's value for [2]

live ↗

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

Known value · FindStat St001593

FindStat's value for [3]

live ↗

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

Known value · FindStat St001593

About

Examples	5
Crosswalk	FindStat