Identifier
Values
0 => 1 => ([(0,1)],2) => 2
1 => 0 => ([(0,1)],2) => 2
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
01 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
10 => 11 => ([(0,2),(2,1)],3) => 3
11 => 00 => ([(0,2),(2,1)],3) => 3
000 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
001 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 7
010 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 7
011 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
100 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
101 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
110 => 111 => ([(0,3),(2,1),(3,2)],4) => 4
111 => 000 => ([(0,3),(2,1),(3,2)],4) => 4
1110 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
1111 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
11111 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
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Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Map
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.
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 $v$.