- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001172: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 0

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

[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0

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

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

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

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

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

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

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

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

[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 2

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

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

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

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

[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 1

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

[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 1

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

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

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

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

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

[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 1

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

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

[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 1

[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0

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

[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 1

[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 1

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

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

[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0] => 1

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

[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0] => 2

[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0] => 2

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

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

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

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

[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0] => 0

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

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

[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0] => 2

[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0] => 1

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

[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => 0

[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => 1

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

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

[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0] => 1

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

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

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

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

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

[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0] => 0

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

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

>>> Load all 211 entries. <<<

[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,1,0,1,0,0] => 2

[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => 1

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

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

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

[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0] => 1

[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,1,0,1,0,0,0] => 0

[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0] => 1

[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0] => 1

[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 0

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

[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0] => 0

[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => 1

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

[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,1,0,1,0,0,0] => 0

[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0] => 1

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

[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 0

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

[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0] => 1

[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0] => 1

[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 0

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

[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0

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

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

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

[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0] => 2

[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => 2

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

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

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

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

[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0] => 1

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

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

[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0] => 1

[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0] => 0

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

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

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

[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0] => 1

[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0] => 1

[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0] => 2

[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => 2

[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0] => 1

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

[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0] => 1

[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1

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

[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0] => 0

[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0] => 1

[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0] => 2

[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0] => 1

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

[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0] => 0

[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0] => 1

[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0] => 1

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

[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0] => 1

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

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

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

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

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

[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0] => 0

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

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

[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0] => 2

[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0] => 1

[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0] => 1

[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0] => 0

[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0] => 1

[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0] => 1

[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0] => 0

[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0] => 1

[2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1

[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

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

[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0] => 1

[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,1,0,0,0,1,0,0] => 1

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

[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0] => 0

[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0] => 1

[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0] => 2

[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0] => 1

[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0] => 1

[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0] => 0

[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0] => 1

[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0] => 1

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

[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0] => 1

[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0] => 1

[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 0

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

[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0] => 0

[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0] => 1

[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0] => 1

[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0] => 0

[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0] => 1

[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0] => 1

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

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

[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0] => 1

[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0] => 1

[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 0

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

[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + q$

$F_{4} = 5 + 3\ q$

$F_{5} = 8 + 8\ q$

$F_{6} = 14 + 17\ q + q^{2}$

$F_{7} = 24 + 36\ q + 4\ q^{2}$

Description

The number of 1-rises at odd height of a Dyck path.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

Map

inverse promotion

Description

The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.