Identifier
Values
0 => 0 => ([(0,1)],2) => ([(0,1)],2) => 0
1 => 1 => ([(0,1)],2) => ([(0,1)],2) => 0
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
01 => 00 => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => 0
10 => 11 => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => 0
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
000 => 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) => 1
001 => 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) => 0
010 => 000 => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => 0
011 => 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) => 0
100 => 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) => 0
101 => 111 => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => 0
110 => 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) => 0
111 => 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) => 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 0
10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 0
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) => 0
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) => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph.
A graph is outerplanar if and only if in any linear ordering of its vertices, there are no four vertices $a < b < c < d$ such that $(a,c)$ and $(b,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
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.