ElementarySymmetricFunctionAtOne

The elementary symmetric function indexed by the partition, in as many variables as it has parts, all set to 1: the product of the binomial coefficients of the number of parts and each part.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001561via IntegerPartition✓↗

ElementarySymmetricFunctionAtOne(partition)The elementary symmetric function indexed by the partition, in as many variables as it has parts, all set to 1: the product of the binomial coefficients of the number of parts and each part.

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

Examples

Basic #

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

live ↗

In$\mathrm{ElementarySymmetricFunctionAtOne}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$0$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001561

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{ElementarySymmetricFunctionAtOne})$Out$0$

Known value · FindStat St001561

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{ElementarySymmetricFunctionAtOne})$Out$0$

Known value · FindStat St001561

About

Examples	5
Crosswalk	FindStat