- FindStat

Identifier

Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000790: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path St000012The area of a Dyck path..

Map

Barnabei-Castronuovo involution

Description

The Barnabei-Castronuovo Schützenberger involution on Dyck paths.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let $D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is $\dots, 1, D_2, 1, D_1$.