- FindStat

Identifier

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001688: Dyck paths ⟶ ℤ

Values

[1,0] => [1,0] => 1

[1,0,1,0] => [1,0,1,0] => 2

[1,1,0,0] => [1,1,0,0] => 4

[1,0,1,0,1,0] => [1,0,1,0,1,0] => 3

[1,0,1,1,0,0] => [1,0,1,1,0,0] => 5

[1,1,0,0,1,0] => [1,1,0,0,1,0] => 5

[1,1,0,1,0,0] => [1,0,1,1,0,0] => 5

[1,1,1,0,0,0] => [1,1,1,0,0,0] => 9

[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 4

[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => 6

[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 6

[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 6

[1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 10

[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 6

[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 8

[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 6

[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 8

[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 10

[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 10

[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 8

[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 10

[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 16

[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 7

[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 7

[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 7

[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 11

[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 7

[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9

[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 7

[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9

[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 11

[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 11

[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9

[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 11

[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 17

[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 7

[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 9

[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 9

[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 9

[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 13

[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 7

[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9

[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 9

[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9

[1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 13

[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 11

[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9

[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 13

[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 17

[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 11

[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 13

[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 9

[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 13

[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 13

[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 11

[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 13

[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 13

[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 17

[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 17

[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 13

[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 13

[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 17

[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 25

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 8

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 8

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 8

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 12

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 8

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 8

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 12

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 12

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 12

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 18

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 8

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 10

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 14

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 8

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 10

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 14

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 12

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 14

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 18

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 12

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 14

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 10

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 14

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 14

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 12

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 14

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 14

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 18

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Description

The sum of the squares of the heights of the peaks of a Dyck path.

Map

bounce path

Description

Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.