CompleteHomogeneousSymmetricFunctionAtOne

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

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001562via IntegerPartition✓↗

CompleteHomogeneousSymmetricFunctionAtOne(partition)The complete homogeneous symmetric function indexed by the partition, in as many variables as it has parts, all set to 1: the product of the multiset 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) - 1 +
                                                           _x[i] else (1 if Length(_x) -
                                                                       1 + _x[i] ==
                                                                       0 else Product(Range(1, Length(_x) - 1 +
                                                                                              _x[i]))) /
                                                                 ((1 if _x[i] ==
                                                                        0 else Product(Range(1, _x[i]))) *
                                                                  (1 if Length(_x) -
                                                                        1 + _x[i] -
                                                                        _x[i] == 0 else Product(Range(1, Length(_x) - 1 +
                                                                                                    _x[i] -
                                                                                                    _x[i])))), i), Range(1, Length(_x))))

Examples

Basic #

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

live ↗

In$\mathrm{CompleteHomogeneousSymmetricFunctionAtOne}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$270$

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

live ↗

In$\mathrm{CompleteHomogeneousSymmetricFunctionAtOne}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$300$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001562

FindStat's value for [2]

live ↗

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

Known value · FindStat St001562

FindStat's value for [3]

live ↗

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

Known value · FindStat St001562

About

Examples	5
Crosswalk	FindStat