CyclicSievingPolynomialDegree

(1 - 1/largest part) times the least common multiple of the parts: the degree of the cyclic sieving polynomial of a permutation with these cycle lengths.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001918via IntegerPartition✓↗

CyclicSievingPolynomialDegree(partition)(1 - 1/largest part) times the least common multiple of the parts: the degree of the cyclic sieving polynomial of a permutation with these cycle lengths.

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.
• FindStat leaves the empty partition undefined; it is 0 here.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else (Max(_x) - 1) * Fold(
                                               Function(LCM(a, b), a, b),
                                               _x[1],
                                               Rest(_x),
                                             ) / Max(_x)

Examples

Basic #

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

live ↗

In$\mathrm{CyclicSievingPolynomialDegree}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$3$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{CyclicSievingPolynomialDegree})$Out$0$

Known value · FindStat St001918

FindStat's value for [2]

live ↗

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

Known value · FindStat St001918

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{CyclicSievingPolynomialDegree})$Out$2$

Known value · FindStat St001918

About

Examples	5
Crosswalk	FindStat