Finset

The singular-inhabitant constructor for a subset of $\{1, \dots, n\}$ , as $n$ and its members.

Finset(tuple)The singular-inhabitant constructor for a subset of

\{1, \dots, n\}

, as

n

and its members.

Typed signatures

Type	Package	Applies
`((tuple<integer, list<integer>>) -> finset) & ((binary_word) -> finset)`	`combinatorics`	always

Domain: Collections

Details

A Subsets element: Finset(Tuple(n, members)), the members ascending.
Finset(word) reads a characteristic word: the positions holding 1. Over BinaryWords, that lists the subsets of $\{1, \dots, n\}$ in their words' order, where Subsets lists them by size, then lex.

Maps

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

BinaryWordFinset → BinaryWordinverseA subset of 1..n as its characteristic word: letter i is 1 when i is a member.

DescentSetPermutation → FinsetThe positions where the word falls.
FinsetBinaryWord → FinsetinverseThe subset of 1..m a binary word of length m marks: the positions holding 1.
PeakSetPermutation → FinsetThe interior positions that rise then fall.
PermutahedronVertexPermutation → FinsetThe vertex (σ(1), …, σ(n)), or its Loday point, is an ordered tuple of coordinates, while finset is an unordered set of positions (the carrier DescentSet and PeakSet use). Reordering the coordinates gives a different vertex but the same finset, so the map would not be injective. A finset-valued PermutahedronVertex is a type mismatch, not a gap to close.

Basic #

From its characteristic word, the inverse of BinaryWord on a subset

\operatorname{Finset}({\operatorname{BinaryWord}}([0, 1, 1, 0, 1]))

Out

\operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack2, 3, 5\bigr\rbrack))

About

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