- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001213: Dyck paths ⟶ ℤ

Values

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

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

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

[1,0,1,0,1,0] => [3,2,1] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 6

[1,0,1,1,0,0] => [2,3,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 5

[1,1,0,0,1,0] => [3,1,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 6

[1,1,0,1,0,0] => [2,1,3] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 5

[1,1,1,0,0,0] => [1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 4

[1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 8

[1,0,1,0,1,1,0,0] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 7

[1,0,1,1,0,0,1,0] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 8

[1,0,1,1,0,1,0,0] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 7

[1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 6

[1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 8

[1,1,0,0,1,1,0,0] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 7

[1,1,0,1,0,0,1,0] => [4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 8

[1,1,0,1,0,1,0,0] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 7

[1,1,0,1,1,0,0,0] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 6

[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 8

[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 7

[1,1,1,0,1,0,0,0] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 6

[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 5

[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 9

[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 8

[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 9

[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 8

[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 9

[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 8

[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 7

[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 9

[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 8

[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 8

[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 8

[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 7

[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 8

[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 8

[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 7

[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 8

[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 7

[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 6

[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11

[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 11

[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 11

[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 10

[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 9

[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11

[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 11

[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 11

[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 10

[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 9

[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 11

[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 11

[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10

[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 9

>>> Load all 196 entries. <<<

[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 11

[1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10

[1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 9

[1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 8

[1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11

[1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,0,1,1,0,1,0,1,0,0] => [5,4,3,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 11

[1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 11

[1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 10

[1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 9

[1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11

[1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,1,0,1,0,1,0,0,1,0,1,0] => [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,0,1,0,1,0,1,1,0,0,0] => [4,5,3,2,1,6] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10

[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 9

[1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,0,1,1,0,0,1,1,0,0,0] => [4,5,2,3,1,6] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,5,1,6] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 10

[1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 9

[1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,3,5,1,6] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 10

[1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,4,5,1,6] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 9

[1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 8

[1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11

[1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,1,1,0,0,1,0,0,1,0,1,0] => [6,5,3,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,0,0,1,0,1,1,0,0,0] => [4,5,3,1,2,6] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,0,1,1,0,0,1,0,0] => [5,3,4,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,0,0,1,1,0,1,0,0,0] => [4,3,5,1,2,6] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10

[1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 9

[1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,0,1,0,0,1,1,0,0,0] => [4,5,2,1,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 10

[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 9

[1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 10

[1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,4,1,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 9

[1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 8

[1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,1,0,0,0,1,1,0,0,0] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 10

[1,1,1,1,0,0,1,1,0,0,0,0] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 9

[1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 10

[1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9

[1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 8

[1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 12

[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 11

[1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 10

[1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9

[1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 8

[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 7

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$F_{1} = q^{2}$

$F_{2} = q^{3} + q^{4}$

$F_{3} = q^{4} + 2\ q^{5} + 2\ q^{6}$

$F_{4} = q^{5} + 3\ q^{6} + 5\ q^{7} + 5\ q^{8}$

$F_{5} = q^{6} + 4\ q^{7} + 9\ q^{8} + 14\ q^{9} + 14\ q^{10}$

$F_{6} = q^{7} + 5\ q^{8} + 14\ q^{9} + 28\ q^{10} + 42\ q^{11} + 42\ q^{12}$

Description

The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].