Identifier
Values
[1] => [1,0] => 1
[1,1] => [1,0,1,0] => 1
[2] => [1,1,0,0] => 2
[1,1,1] => [1,0,1,0,1,0] => 1
[1,2] => [1,0,1,1,0,0] => 2
[2,1] => [1,1,0,0,1,0] => 2
[3] => [1,1,1,0,0,0] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => 2
[1,3] => [1,0,1,1,1,0,0,0] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => 2
[2,2] => [1,1,0,0,1,1,0,0] => 2
[3,1] => [1,1,1,0,0,0,1,0] => 3
[4] => [1,1,1,1,0,0,0,0] => 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 4
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
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Description
The difference of lower and upper interactions.
An upper interaction in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see St000331The number of upper interactions of a Dyck path.), and a lower interaction is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Map
bounce path
Description
The bounce path determined by an integer composition.