- FindStat

Identifier

Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001656: Graphs ⟶ ℤ

Values

=> 0 => ([(0,1)],2) => ([(0,1)],2) => 2
=> 1 => ([(0,1)],2) => ([(0,1)],2) => 2
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
=> 00 => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => 2
=> 11 => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => 2
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
=> 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 2
=> 000 => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => 2
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 2
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 2
=> 111 => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => 2
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 2
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
=> 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
010101 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 2
101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 2

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$F_{1} = 2\ q^{2}$

$F_{2} = 4\ q^{2}$

$F_{3} = 6\ q^{2} + 2\ q^{3}$

Description

The monophonic position number of a graph.
A subset

$M$ of the vertex set of a graph is a monophonic position set if no three vertices of

$M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.

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

to graph

Description

Returns the Hasse diagram of the poset as an undirected graph.

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.