Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000681
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [2,2]
 => 2
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [3]
 => 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [2,2,2]
 => [2,2]
 => 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [2]
 => 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
 => [3,3]
 => [3]
 => 2
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 3
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => [2,1,1,1]
 => [1,1,1]
 => 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [2,2,1,1]
 => [2,1,1]
 => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [3,1,1]
 => [1,1]
 => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [2,2,2,1]
 => [2,2,1]
 => 0
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 0
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [3,3,1]
 => [3,1]
 => 3
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => [2,2,2]
 => 4
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => [2,2]
 => 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => [3,2,2]
 => [2,2]
 => 2
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [2]
 => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [3,3,2]
 => [3,2]
 => 0
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 1
[3,3,1] => [[5,5,3],[4,2]]
 => [4,2]
 => [2]
 => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => [3,3]
 => 4
[4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => [3]
 => 2
[4,2,1] => [[5,5,4],[4,3]]
 => [4,3]
 => [3]
 => 2
[5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => [4]
 => 3
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
 => [2,1,1]
 => [1,1]
 => 1
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 0
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 1
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001364
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001364: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0 = 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]
 => [4,3]
 => 0 = 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]
 => [5,4,3,1]
 => ? = 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]
 => [5,4,1]
 => ? = 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]
 => [5,4,3,2]
 => ? = 2 - 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]
 => [4,3,2]
 => 0 = 1 - 1
[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]
 => [5,3,2]
 => ? = 1 - 1
[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]
 => [5,4,2]
 => ? = 0 - 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]
 => [5,4,3]
 => ? = 2 - 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]
 => [4,3]
 => 0 = 1 - 1
[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]
 => [5,3]
 => 0 = 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]
 => [5,4]
 => 1 = 2 - 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]
 => [6,5,4,2,1]
 => ? = 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]
 => [6,5,2,1]
 => ? = 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]
 => [6,5,4,3,1]
 => ? = 2 - 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]
 => [5,4,3,1]
 => ? = 1 - 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1]
 => ? = 1 - 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1]
 => ? = 0 - 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1]
 => ? = 2 - 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]
 => [5,4,1]
 => ? = 1 - 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [6,4,1]
 => ? = 1 - 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]
 => [6,5,1]
 => ? = 2 - 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => ? = 3 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 2 - 1
[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]
 => [6,4,3,2]
 => ? = 2 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[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]
 => [6,5,3,2]
 => ? = 3 - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [5,3,2]
 => ? = 1 - 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [6,3,2]
 => ? = 1 - 1
[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]
 => [6,5,4,2]
 => ? = 0 - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [5,4,2]
 => ? = 0 - 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [6,4,2]
 => ? = 0 - 1
[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]
 => [6,5,2]
 => ? = 3 - 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3]
 => ? = 4 - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [5,4,3]
 => ? = 2 - 1
[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]
 => [6,4,3]
 => ? = 2 - 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [4,3]
 => 0 = 1 - 1
[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]
 => [6,5,3]
 => ? = 0 - 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [5,3]
 => 0 = 1 - 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [6,3]
 => 0 = 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]
 => [6,5,4]
 => ? = 4 - 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,4]
 => 1 = 2 - 1
[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]
 => [6,4]
 => ? = 2 - 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 3 - 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1]
 => ? = 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]
 => [7,6,5,4,2,1]
 => ? = 2 - 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => [7,5,4,2,1]
 => ? = 1 - 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => [7,6,4,2,1]
 => ? = 0 - 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]
 => [6,5,2,1]
 => ? = 1 - 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1]
 => ? = 1 - 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]
 => [7,6,5,4,3,1]
 => ? = 3 - 1
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,1]
 => ? = 2 - 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,1]
 => ? = 2 - 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1]
 => ? = 3 - 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3,1]
 => ? = 1 - 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1]
 => ? = 1 - 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1]
 => ? = 0 - 1
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3,1]
 => ? = 0 - 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1]
 => ? = 0 - 1
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1]
 => ? = 3 - 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => [5,3]
 => 0 = 1 - 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => [6,3]
 => 0 = 1 - 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]
 => [5,4]
 => 1 = 2 - 1
Description
The number of permutations whose cube equals a fixed permutation of given cycle type. For example, the permutation $\pi=412365$ has cycle type $(4,2)$ and $234165$ is the unique permutation whose cube is $\pi$.
Matching statistic: St001186
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 30%
Values
[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
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 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
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 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]
 => 2
[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
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 2
[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
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 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]
 => ? = 2
[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
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 2
[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]
 => ? = 3
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 0
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 0
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => ? = 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => ? = 0
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => ? = 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => ? = 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => ? = 2
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 0
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 3
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 0
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 0
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
Description
Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St000365
(load all 3 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 30%
Values
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,1,4,5,2] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,1,2,5,3] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [3,4,2,5,6,1] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [4,5,2,3,6,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [3,1,4,5,6,2] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [3,1,4,6,5,2] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [3,1,5,6,4,2] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,1,2] => [4,1,5,3,6,2] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1,2,3] => [4,1,2,5,6,3] => 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [5,6,4,1,2,3] => [4,1,2,6,5,3] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [6,4,5,1,2,3] => [5,1,2,6,4,3] => 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => [5,1,2,3,6,4] => 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [2,4,5,3,6,7,1] => ? = 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [2,5,6,3,4,7,1] => ? = 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [3,4,2,5,6,7,1] => ? = 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [3,4,2,5,7,6,1] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [3,4,2,6,7,5,1] => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [3,5,2,6,4,7,1] => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => [4,5,2,3,6,7,1] => ? = 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => [4,5,2,3,7,6,1] => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => [4,6,2,3,7,5,1] => ? = 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,5,1] => [5,6,2,3,4,7,1] => ? = 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [3,1,4,5,6,7,2] => ? = 3
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => [3,1,4,5,7,6,2] => ? = 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => [3,1,4,6,7,5,2] => ? = 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [3,1,4,7,5,6,2] => ? = 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => [3,1,5,6,4,7,2] => ? = 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => [3,1,5,7,4,6,2] => ? = 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => [3,1,6,7,4,5,2] => ? = 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,1,2] => [4,1,5,3,6,7,2] => ? = 0
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => [4,1,5,3,7,6,2] => ? = 0
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,1,2] => [4,1,6,3,7,5,2] => ? = 0
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,1,2] => [5,1,6,3,4,7,2] => ? = 3
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => [4,1,2,5,6,7,3] => ? = 4
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [4,1,2,5,7,6,3] => ? = 2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,1,2,3] => [4,1,2,6,7,5,3] => ? = 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,1,2,3] => [4,1,2,7,5,6,3] => ? = 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,1,2,3] => [5,1,2,6,4,7,3] => ? = 0
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,1,2,3] => [5,1,2,7,4,6,3] => ? = 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,1,2,3] => [6,1,2,7,4,5,3] => ? = 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1,2,3,4] => [5,1,2,3,6,7,4] => ? = 4
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,7,5,1,2,3,4] => [5,1,2,3,7,6,4] => ? = 2
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [7,5,6,1,2,3,4] => [6,1,2,3,7,5,4] => ? = 2
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => ? = 3
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,5,3,2,1] => [2,3,5,6,4,7,8,1] => ? = 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,3,4,2,1] => [2,4,5,3,6,7,8,1] => ? = 2
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,4,2,1] => [2,4,5,3,6,8,7,1] => ? = 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,3,4,2,1] => [2,4,5,3,7,8,6,1] => ? = 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,3,4,2,1] => [2,4,6,3,7,5,8,1] => ? = 0
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,5,2,1] => [2,5,6,3,4,8,7,1] => ? = 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [8,6,7,3,4,5,2,1] => [2,5,7,3,4,8,6,1] => ? = 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,2,3,1] => [3,4,2,5,6,7,8,1] => ? = 3
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,2,3,1] => [3,4,2,5,6,8,7,1] => ? = 2
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,4,2,3,1] => [3,4,2,5,7,8,6,1] => ? = 2
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,4,2,3,1] => [3,4,2,6,7,5,8,1] => ? = 3
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,2,3,1] => [3,4,2,6,8,5,7,1] => ? = 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [8,5,6,7,4,2,3,1] => [3,4,2,7,8,5,6,1] => ? = 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,5,2,3,1] => [3,5,2,6,4,7,8,1] => ? = 0
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,2,3,1] => [3,5,2,6,4,8,7,1] => ? = 0
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [8,6,7,4,5,2,3,1] => [3,5,2,7,4,8,6,1] => ? = 0
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [8,7,4,5,6,2,3,1] => [3,6,2,7,4,5,8,1] => ? = 3
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,2,3,4,1] => ? => ? = 2
Description
The number of double ascents of a permutation. A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St001294
Mapping
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 30%
Values
[2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,1,1,1] => [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]
 => 2 = 1 + 1
[1,3,1,1] => [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]
 => 2 = 1 + 1
[2,1,1,1,1] => [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]
 => 3 = 2 + 1
[2,1,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,2,1] => [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]
 => 2 = 1 + 1
[2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[3,1,1,1] => [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]
 => 3 = 2 + 1
[3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [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]
 => 2 = 1 + 1
[4,1,1] => [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]
 => 3 = 2 + 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]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 1
[1,1,3,1,1] => [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 + 1
[1,2,1,1,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,2,1,1,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 1
[1,2,1,2,1] => [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 + 1
[1,2,2,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[1,3,1,1,1] => [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]
 => ? = 2 + 1
[1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,3,2,1] => [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 + 1
[1,4,1,1] => [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]
 => ? = 2 + 1
[2,1,1,1,1,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,1,1,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[2,1,1,2,1] => [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]
 => ? = 2 + 1
[2,1,1,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 1
[2,1,2,1,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,2,2] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[2,1,3,1] => [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 + 1
[2,2,1,1,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 0 + 1
[2,2,1,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[2,2,2,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0 + 1
[2,3,1,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[3,1,1,1,1] => [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]
 => ? = 4 + 1
[3,1,1,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 2 + 1
[3,1,2,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 2 + 1
[3,1,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[3,2,1,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[3,3,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[4,1,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 4 + 1
[4,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[4,2,1] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,1,1,2,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 1
[1,1,2,1,1,1,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,1,2,1,1,2] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 1
[1,1,2,1,2,1] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,1,2,2,1,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[1,1,3,1,2] => [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
[1,1,3,2,1] => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,1,1,1,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,2,1,1,1,2] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,2,1,1,2,1] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 2 + 1
[1,2,1,2,1,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 3 + 1
[1,2,1,2,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,2,1,3,1] => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,2,2,1,1,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 0 + 1
[1,2,2,1,2] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[1,2,2,2,1] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0 + 1
[1,2,3,1,1] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,3,1,1,2] => [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]
 => ? = 2 + 1
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]]. The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.