CombinatorialStat

A combinatorial statistic of a value, by name or FindStat id, found through the value's carrier.

CombinatorialStat(x, name)the statistic name (or FindStat id, "St000018") of x, a value of a carrier such as Permutation; over a collection, the statistic of each element, lazily, and "Count" the collection's size.

Typed signatures

Type	Package	Applies
(any, any) -> any	structures	always

Domain: Structures

Details

• A carrier's statistics are too many, and too generically named, for a head each: they live in the carrier's table, and this is the way in. Packages add to a table, a fast kernel from one and the defining expression from another; the same part twice is an error.
• The carrier is found from the value's type, as a protocol member finds its implementation, so a type a user declares can have statistics too.
• Over a collection it is the statistic's distribution: Tally(CombinatorialStat(SymmetricGroup(4), "Inversions")) is the Mahonian numbers.
• Where FindStat has the statistic, its id is another name for it. Several of our names can be one FindStat statistic.

Examples

Basic #

the inversions of 312

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([3, 1, 2]), \text{Inversions})$Out$2$

the same statistic, by its FindStat id

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([3, 1, 2]), \text{St000018})$Out$2$

a statistic whose bare name is compute-engine's Sign

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([3, 1, 2]), \text{Sign})$Out$1$

the distinct parts of 311

live ↗

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

a statistic of the collection itself

live ↗

In$\mathrm{CombinatorialStat}(\mathrm{SymmetricGroup}(4), \text{Count})$Out$24$

Applications #

inversions over the permutations of 4

live ↗

In$\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{SymmetricGroup}(4), \text{Inversions}))$Out$\operatorname{Tuple}(\bigl\lbrack0, 1, 2, 3, 4, 5, 6\bigr\rbrack,\bigl\lbrack1, 3, 5, 6, 5, 3, 1\bigr\rbrack)$

About

Examples	6
Checked against	SageMath 2 agree

See also: CombinatorialMap, Count