Your data matches 168 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
(load all 6 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,4,1] => [[4,4,1,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,4] => [[5,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001033
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [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
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [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
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [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
[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]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [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
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 3
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [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
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [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
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => 3
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => 3
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
 => ? = 5
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
 => ? = 4
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
 => ? = 3
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => ? = 3
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
Description
The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St000147
Mapping
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 1
[1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 1
[2,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 2
[2,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,1] => [2,3] => [[4,2],[1]]
 => [1]
 => 1
[1,2,1,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 2
[1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[2,1,1,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 3
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2
[2,3] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
[3,1,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[3,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[4,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,1,1,2,1] => [2,4] => [[5,2],[1]]
 => [1]
 => 1
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
 => [2]
 => 2
[1,1,2,2] => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1
[1,1,3,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
[1,2,1,1,1] => [4,2] => [[5,4],[3]]
 => [3]
 => 3
[1,2,1,2] => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 2
[1,2,3] => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
[1,3,2] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,1,1,1] => [5,1] => [[5,5],[4]]
 => [4]
 => 4
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 3
[2,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 2
[2,4] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[3,1,1,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[3,1,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
[4,1,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
[4,2] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
[1,1,1,1,2,1] => [2,5] => [[6,2],[1]]
 => [1]
 => 1
[1,1,1,2,1,1] => [3,4] => [[6,3],[2]]
 => [2]
 => 2
[1,1,1,2,2] => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => 1
[1,1,1,3,1] => [2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,1,1,1] => [4,3] => [[6,4],[3]]
 => [3]
 => 3
[1,1,2,1,2] => [1,3,3] => [[5,3,1],[2]]
 => [2]
 => 2
[1,1,2,3] => [1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 1
[1,1,3,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 2
[1,1,3,2] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,1,4,1] => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,2,1,1,1,1] => [5,2] => [[6,5],[4]]
 => [4]
 => 4
[1,2,1,1,2] => [1,4,2] => [[5,4,1],[3]]
 => [3]
 => 3
[1,2,1,3] => [1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => 2
[1,2,4] => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 1
[1,3,1,1,1] => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 3
[1,1,1,1,1,2,1] => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
[1,1,1,1,2,1,1] => [3,5] => [[7,3],[2]]
 => ?
 => ? = 2
[1,1,1,1,2,2] => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1
[1,1,1,2,1,2] => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 2
[1,1,1,3,2] => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => [5,3] => [[7,5],[4]]
 => ?
 => ? = 4
[1,1,3,3] => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => [6,2] => [[7,6],[5]]
 => ?
 => ? = 5
[1,2,1,1,1,2] => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 4
[2,1,1,1,1,2] => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 5
[2,1,1,3,1] => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 5
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 3
[4,1,3] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 2
Description
The largest part of an integer partition.
Matching statistic: St001392
Mapping
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0 = 1 - 1
[2,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 1 = 2 - 1
[2,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,2,1] => [2,3] => [[4,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,2,1,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[2,1,1,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2 = 3 - 1
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1 = 2 - 1
[2,3] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[3,1,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[3,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[4,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,1,2,1] => [2,4] => [[5,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,2,2] => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,3,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,2,1,1,1] => [4,2] => [[5,4],[3]]
 => [3]
 => 2 = 3 - 1
[1,2,1,2] => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1 = 2 - 1
[1,2,3] => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[1,3,2] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[2,1,1,1,1] => [5,1] => [[5,5],[4]]
 => [4]
 => 3 = 4 - 1
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 2 = 3 - 1
[2,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1 = 2 - 1
[2,4] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[3,1,1,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2 = 3 - 1
[3,1,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[4,1,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 1 = 2 - 1
[4,2] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,1,1,1,2,1] => [2,5] => [[6,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,2,1,1] => [3,4] => [[6,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,1,2,2] => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,3,1] => [2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,2,1,1,1] => [4,3] => [[6,4],[3]]
 => [3]
 => 2 = 3 - 1
[1,1,2,1,2] => [1,3,3] => [[5,3,1],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,2,3] => [1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,3,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[1,1,3,2] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,4,1] => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[1,2,1,1,1,1] => [5,2] => [[6,5],[4]]
 => [4]
 => 3 = 4 - 1
[1,2,1,1,2] => [1,4,2] => [[5,4,1],[3]]
 => [3]
 => 2 = 3 - 1
[1,2,1,3] => [1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => 1 = 2 - 1
[1,2,4] => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,3,1,1,1] => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 2 = 3 - 1
[1,1,1,1,1,2,1] => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1 - 1
[1,1,1,1,2,1,1] => [3,5] => [[7,3],[2]]
 => ?
 => ? = 2 - 1
[1,1,1,1,2,2] => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1 - 1
[1,1,1,2,1,2] => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 2 - 1
[1,1,1,3,2] => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 1 - 1
[1,1,2,1,1,1,1] => [5,3] => [[7,5],[4]]
 => ?
 => ? = 4 - 1
[1,1,3,3] => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
 => ?
 => ? = 1 - 1
[1,2,1,1,1,1,1] => [6,2] => [[7,6],[5]]
 => ?
 => ? = 5 - 1
[1,2,1,1,1,2] => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 4 - 1
[2,1,1,1,1,2] => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 5 - 1
[2,1,1,3,1] => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 5 - 1
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 3 - 1
[4,1,3] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 2 - 1
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001918
Mapping
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => 0 = 1 - 1
[2,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 1 = 2 - 1
[2,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,2,1] => [2,3] => [[4,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,2,1,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[2,1,1,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 2 = 3 - 1
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1 = 2 - 1
[2,3] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[3,1,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[3,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[4,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,1,2,1] => [2,4] => [[5,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,2,2] => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,3,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,2,1,1,1] => [4,2] => [[5,4],[3]]
 => [3]
 => 2 = 3 - 1
[1,2,1,2] => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1 = 2 - 1
[1,2,3] => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[1,3,2] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[2,1,1,1,1] => [5,1] => [[5,5],[4]]
 => [4]
 => 3 = 4 - 1
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 2 = 3 - 1
[2,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1 = 2 - 1
[2,4] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[3,1,1,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2 = 3 - 1
[3,1,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[4,1,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 1 = 2 - 1
[4,2] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,1,1,1,2,1] => [2,5] => [[6,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,2,1,1] => [3,4] => [[6,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,1,2,2] => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,3,1] => [2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,2,1,1,1] => [4,3] => [[6,4],[3]]
 => [3]
 => 2 = 3 - 1
[1,1,2,1,2] => [1,3,3] => [[5,3,1],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,2,3] => [1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,3,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 1 = 2 - 1
[1,1,3,2] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,4,1] => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[1,2,1,1,1,1] => [5,2] => [[6,5],[4]]
 => [4]
 => 3 = 4 - 1
[1,2,1,1,2] => [1,4,2] => [[5,4,1],[3]]
 => [3]
 => 2 = 3 - 1
[1,2,1,3] => [1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => 1 = 2 - 1
[1,2,4] => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,3,1,1,1] => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 2 = 3 - 1
[1,1,1,1,1,2,1] => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1 - 1
[1,1,1,1,2,1,1] => [3,5] => [[7,3],[2]]
 => ?
 => ? = 2 - 1
[1,1,1,1,2,2] => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1 - 1
[1,1,1,2,1,2] => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 2 - 1
[1,1,1,3,2] => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 1 - 1
[1,1,2,1,1,1,1] => [5,3] => [[7,5],[4]]
 => ?
 => ? = 4 - 1
[1,1,3,3] => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
 => ?
 => ? = 1 - 1
[1,2,1,1,1,1,1] => [6,2] => [[7,6],[5]]
 => ?
 => ? = 5 - 1
[1,2,1,1,1,2] => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 4 - 1
[2,1,1,1,1,2] => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 5 - 1
[2,1,1,3,1] => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 5 - 1
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 3 - 1
[4,1,3] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 2 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 2 = 1 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 4 + 1
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4 + 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 2 + 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 1 + 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001000
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%
Values
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,4,1] => [[4,4,1,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,2,4] => [[5,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
[6,1] => [[6,6],[5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[4,1,1,2] => [[5,4,4,4],[3,3,3]]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[6,1,1] => [[6,6,6],[5,5]]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
[7,1] => [[7,7],[6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000052
(load all 4 compositions to match this statistic)
Mapping
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 1
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 3
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 2
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3
[3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[4,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,1,1,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,2,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,1,2,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,3,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => 1
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,2,1,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => 1
[1,1,3,1,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => 2
[1,1,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => 1
[1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => 1
[1,2,1,1,1,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[1,2,1,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => 3
[1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => 2
[1,2,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => 1
[1,3,1,1,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => 3
[1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,2,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,2,1,2] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,3,2] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,3,1,2] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 1
[1,2,1,3,1] => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 3
[1,3,1,1,2] => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 3
[1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 2
[1,4,1,1,1] => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 3
[1,4,1,2] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
[2,1,1,3,1] => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 5
[2,1,3,1,1] => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 4
[2,1,3,2] => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 3
[3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 3
[3,1,3,1] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
 => ? = 3
[4,1,1,1,1] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 4
[4,1,1,2] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000668
Mapping
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [[2,2],[1]]
 => [1]
 => ? = 1
[1,2,1] => [2,2] => [[3,2],[1]]
 => [1]
 => ? = 1
[2,1,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 2
[2,2] => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? = 1
[3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,1] => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 1
[1,2,1,1] => [3,2] => [[4,3],[2]]
 => [2]
 => 2
[1,2,2] => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? = 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[2,1,1,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 3
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2
[2,3] => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? = 1
[3,1,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[3,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[4,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,1,1,2,1] => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
 => [2]
 => 2
[1,1,2,2] => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? = 1
[1,1,3,1] => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
[1,2,1,1,1] => [4,2] => [[5,4],[3]]
 => [3]
 => 3
[1,2,1,2] => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 2
[1,2,3] => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? = 1
[1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
[1,3,2] => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,1,1,1] => [5,1] => [[5,5],[4]]
 => [4]
 => 4
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 3
[2,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 2
[2,4] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => ? = 1
[3,1,1,1] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[3,1,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
[4,1,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
[4,2] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
[1,1,1,1,2,1] => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
[1,1,1,2,1,1] => [3,4] => [[6,3],[2]]
 => [2]
 => 2
[1,1,1,2,2] => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => ? = 1
[1,1,1,3,1] => [2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,2,1,1,1] => [4,3] => [[6,4],[3]]
 => [3]
 => 3
[1,1,2,1,2] => [1,3,3] => [[5,3,1],[2]]
 => [2]
 => 2
[1,1,2,3] => [1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => ? = 1
[1,1,3,1,1] => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 2
[1,1,3,2] => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,1,4,1] => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,2,1,1,1,1] => [5,2] => [[6,5],[4]]
 => [4]
 => 4
[1,2,1,1,2] => [1,4,2] => [[5,4,1],[3]]
 => [3]
 => 3
[1,2,1,3] => [1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => 2
[1,2,4] => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => ? = 1
[1,3,1,1,1] => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 3
[1,3,1,2] => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => 2
[1,3,3] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 1
[1,4,1,1] => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
[1,4,2] => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[1,5,1] => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
[2,1,1,1,1,1] => [6,1] => [[6,6],[5]]
 => [5]
 => 5
[2,1,1,1,2] => [1,5,1] => [[5,5,1],[4]]
 => [4]
 => 4
[2,1,1,3] => [1,1,4,1] => [[4,4,1,1],[3]]
 => [3]
 => 3
[2,1,3,1] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => 3
[2,1,4] => [1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 2
[2,5] => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => ? = 1
[3,1,1,1,1] => [5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => 4
[3,1,1,2] => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [3,3]
 => 3
[3,1,3] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => 2
[3,4] => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 1
[1,1,1,1,1,2,1] => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
[1,1,1,1,2,1,1] => [3,5] => [[7,3],[2]]
 => ?
 => ? = 2
[1,1,1,1,2,2] => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1
[1,1,1,2,1,2] => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 2
[1,1,1,3,2] => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => [5,3] => [[7,5],[4]]
 => ?
 => ? = 4
[1,1,3,3] => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => [6,2] => [[7,6],[5]]
 => ?
 => ? = 5
[1,2,1,1,1,2] => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 4
[2,1,1,1,1,2] => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 5
[2,1,1,3,1] => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 5
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 3
[4,1,3] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St001090
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3 = 2 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4 = 3 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 3 = 2 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,2,3,6,4,5] => 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5] => 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,2,5,3,4,6] => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,2,5,3,6,4] => 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,5] => 4 = 3 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,5,2,3,4,6] => 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,4,2,3,6,5] => 2 = 1 + 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]
 => [1,5,2,6,3,4] => 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,2,6,3,5] => 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,6] => 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 5 = 4 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => 4 = 3 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,1,2,3,6,5] => 3 = 2 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4,6] => 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,1,6,2,3,4] => 4 = 3 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,6,2,3,5] => 3 = 2 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4,6] => 2 = 1 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [4,1,5,2,6,3] => 3 = 2 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,5,2,6,4] => 2 = 1 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,4,2,6,5] => 2 = 1 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,2,3,4,7,5,6] => 2 = 1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => 3 = 2 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,2,3,6,4,5,7] => 2 = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,2,3,6,4,7,5] => 2 = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,2,7,3,4,5,6] => 4 = 3 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5,7] => 3 = 2 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,2,5,3,4,7,6] => 2 = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,2,6,3,7,4,5] => 3 = 2 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,2,5,3,7,4,6] => 2 = 1 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,2,5,3,6,4,7] => 2 = 1 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,7,2,3,4,5,6] => 5 = 4 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,5,7] => ? = 3 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,5,2,3,4,7,6] => 3 = 2 + 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,4,2,3,6,5,7] => 2 = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,6,2,7,3,4,5] => 4 = 3 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,5,2,7,3,4,6] => 3 = 2 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,5,2,6,3,7,4] => ? = 2 + 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 2 + 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4,7,6] => ? = 1 + 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,1,7,2,3,4,5] => ? = 4 + 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,7,2,3,4,6] => ? = 3 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4,7,6] => ? = 1 + 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [5,1,6,2,7,3,4] => ? = 3 + 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,1,6,2,7,3,5] => ? = 2 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 1 + 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [4,1,5,2,6,3,7] => ? = 2 + 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,5,2,6,4,7] => ? = 1 + 1
[6,1] => [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]
 => [3,1,4,2,6,5,7] => ? = 1 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,3,4,8,5,6,7] => ? = 2 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,2,3,4,7,5,6,8] => ? = 1 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,2,8,3,4,5,6,7] => ? = 4 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,2,7,3,4,5,6,8] => ? = 3 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
 => [1,2,6,3,8,4,5,7] => ? = 2 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => [1,2,5,3,7,4,6,8] => ? = 1 + 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5,8] => ? = 5 + 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => [6,1,2,3,7,4,8,5] => ? = 4 + 1
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => [5,1,2,3,7,4,8,6] => ? = 3 + 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [7,1,8,2,3,4,5,6] => ? = 5 + 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,7,2,3,4,6,8] => ? = 3 + 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => [5,1,6,2,3,4,8,7] => ? = 3 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [6,1,7,2,8,3,4,5] => ? = 4 + 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [5,1,7,2,8,3,4,6] => ? = 3 + 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,1,6,2,8,3,5,7] => ? = 2 + 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,5,2,7,4,6,8] => ? = 1 + 1
Description
The number of pop-stack-sorts needed to sort a permutation. The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order. A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000451The length of the longest pattern of the form k 1 2. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001368The number of vertices of maximal degree in a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001812The biclique partition number of a graph. St000356The number of occurrences of the pattern 13-2. St000463The number of admissible inversions of a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001323The independence gap of a graph. St001642The Prague dimension of a graph. St000225Difference between largest and smallest parts in a partition. St000617The number of global maxima of a Dyck path. St000145The Dyson rank of a partition. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001083The number of boxed occurrences of 132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000516The number of stretching pairs of a permutation. St001964The interval resolution global dimension of a poset. St000327The number of cover relations in a poset. St000028The number of stack-sorts needed to sort a permutation. St000710The number of big deficiencies of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001727The number of invisible inversions of a permutation. St000358The number of occurrences of the pattern 31-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000741The Colin de Verdière graph invariant. St001535The number of cyclic alignments of a permutation. St000237The number of small exceedances. St000731The number of double exceedences of a permutation. St000956The maximal displacement of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000961The shifted major index of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000286The number of connected components of the complement of a graph. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001691The number of kings in a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000155The number of exceedances (also excedences) of a permutation. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000355The number of occurrences of the pattern 21-3. St000711The number of big exceedences of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000990The first ascent of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001481The minimal height of a peak of a Dyck path. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000039The number of crossings of a permutation. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000732The number of double deficiencies of a permutation. St000798The makl of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001220The width of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001530The depth of a Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001271The competition number of a graph. St001342The number of vertices in the center of a graph. St001305The number of induced cycles on four vertices in a graph. St001316The domatic number of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001571The Cartan determinant of the integer partition. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001435The number of missing boxes in the first row. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St000461The rix statistic of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000989The number of final rises of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001550The number of inversions between exceedances where the greater exceedance is linked. St001622The number of join-irreducible elements of a lattice. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001626The number of maximal proper sublattices of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001438The number of missing boxes of a skew partition.