- FindStat

Identifier

Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000741: Graphs ⟶ ℤ

Values

=> 0 => ([(0,1)],2) => ([],2) => 1
=> 1 => ([(0,1)],2) => ([],2) => 1
=> 00 => ([(0,2),(2,1)],3) => ([],3) => 1
=> 11 => ([(0,2),(2,1)],3) => ([],3) => 1
=> 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 1
=> 000 => ([(0,3),(2,1),(3,2)],4) => ([],4) => 1
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 1
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 1
=> 111 => ([(0,3),(2,1),(3,2)],4) => ([],4) => 1
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 1
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1
=> 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 1
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => 1

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$F_{1} = 2\ q$

Description

The Colin de Verdière graph invariant.

Map

poset of factors

Description

The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that

$u < v$ if and only if

$u$ is a factor of

$v$ .

Map

incomparability graph

Description

The incomparability graph of a poset.

Map

alternating inverse

Description

Sends a binary word

$w_1\cdots w_m$ to the binary word

$v_1 \cdots v_m$ with

$v_i = w_i$ if

$i$ is odd and

$v_i = 1 - w_i$ if

$i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.