RestrictedGrowthString

The singular-inhabitant constructor for a restricted growth string: a word w with w₁ = 0 and each letter at most one more than every letter before it.

WikipediaPartition of a set↗Partition of a set↗MathWorldSetPartition↗WikidataQ381060↗Britannicatopic/partition-of-a-set↗

RestrictedGrowthString(list)The singular-inhabitant constructor for a restricted growth string: a word w with w₁ = 0 and each letter at most one more than every letter before it.

Typed signatures

Type	Package	Applies
((list<integer>) -> restricted_growth_string) & ((set_partition) -> restricted_growth_string)	combinatorics	always

Domain: Collections

Details

• The same structure as a set partition, written as the block index of each position: $01022$ is $\{1,3\} \mid \{2\} \mid \{4,5\}$. The two carriers are joined by an order isomorphism (RestrictedGrowthString(partition), SetPartition(string)), so every statistic and map defined on set partitions answers on restricted growth strings too.

Maps

Into and out of RestrictedGrowthString, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

From RestrictedGrowthString

• SetPartitionRestrictedGrowthString → SetPartitioninverseThe set partition a restricted growth string labels: block j holds the positions labelled j.

To RestrictedGrowthString

• RestrictedGrowthStringSetPartition → RestrictedGrowthStringinverseA set partition's restricted growth string: each position labelled with its block's index, from 0.

Examples

Basic #

From a set partition: each position labelled with its block, from 0

live ↗

In$\operatorname{RestrictedGrowthString}({\operatorname{SetPartition}}([[1, 3], [2], [4]]))$Out${\operatorname{RestrictedGrowthString}}([0, 1, 0, 2])$

About

Examples	1
Crosswalk	Wikipedia ×2MathWorldWikidataBritannica

See also: RestrictedGrowthStrings, SetPartition