- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000937

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: 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]
 => 2
[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]
 => 2
[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]
 => 1
[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]
 => 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [3]
 => 3
[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]
 => 2
[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]
 => 1
[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]
 => 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
 => [3,2]
 => [2]
 => 2
[1,4,1,1] => [[4,4,4,1],[3,3]]
 => [3,3]
 => [3]
 => 3
[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]
 => 2
[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]
 => 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [2,1]
 => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [3,2,1]
 => [2,1]
 => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [3,3,1]
 => [3,1]
 => 2
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => [2,2,2]
 => 5
[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]
 => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [3,3,2]
 => [3,2]
 => 4
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [2]
 => 2
[3,3,1] => [[5,5,3],[4,2]]
 => [4,2]
 => [2]
 => 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => [3,3]
 => 5
[4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => [3]
 => 3
[4,2,1] => [[5,5,4],[4,3]]
 => [4,3]
 => [3]
 => 3
[5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => [4]
 => 5
[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]
 => 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
 => [2,2]
 => [2]
 => 2

Description

The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation

$S^{(2,2)}$ are

$2$ on the conjugacy classes

$(4)$ and

$(2,2)$ ,

$0$ on the conjugacy classes

$(3,1)$ and

$(1,1,1,1)$ , and

$-1$ on the conjugacy class

$(2,1,1)$ . Therefore, the statistic on the partition

$(2,2)$ is

$2$ .

Matching statistic: St000746

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000746: Perfect matchings ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%

Values

[2,1,1,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => 2 = 1 + 1
[3,1,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => ? = 1 + 1
[1,3,1,1] => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => 3 = 2 + 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => ? = 2 + 1
[2,1,1,2] => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 2 = 1 + 1
[2,2,1,1] => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 2 = 1 + 1
[3,1,1,1] => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => 3 = 2 + 1
[3,1,2] => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => 3 = 2 + 1
[3,2,1] => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => 3 = 2 + 1
[4,1,1] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => 4 = 3 + 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
[1,1,3,1,1] => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,2,1,1,1,1] => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,2,1,1,2] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 1 + 1
[1,2,1,2,1] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 1 + 1
[1,2,2,1,1] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 1 + 1
[1,3,1,1,1] => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,3,1,2] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[1,3,2,1] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[1,4,1,1] => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => ? = 3 + 1
[2,1,1,1,1,1] => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => ? = 3 + 1
[2,1,1,1,2] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 2 + 1
[2,1,1,2,1] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 2 + 1
[2,1,1,3] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 1 + 1
[2,1,2,1,1] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 2 + 1
[2,1,2,2] => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 2 = 1 + 1
[2,1,3,1] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 1 + 1
[2,2,1,1,1] => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 3 + 1
[2,2,1,2] => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 2 = 1 + 1
[2,2,2,1] => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 2 = 1 + 1
[2,3,1,1] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[3,1,1,1,1] => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => ? = 5 + 1
[3,1,1,2] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[3,1,2,1] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[3,1,3] => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 3 = 2 + 1
[3,2,1,1] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => ? = 4 + 1
[3,2,2] => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 3 = 2 + 1
[3,3,1] => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 3 = 2 + 1
[4,1,1,1] => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => ? = 5 + 1
[4,1,2] => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => 4 = 3 + 1
[4,2,1] => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => 4 = 3 + 1
[5,1,1] => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => ? = 5 + 1
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,16),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
 => ? = 1 + 1
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,16),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
 => ? = 2 + 1
[1,1,2,1,1,2] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
[1,1,2,1,2,1] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
[1,1,2,2,1,1] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
[1,1,3,1,2] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,1,3,2,1] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,16),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
 => ? = 3 + 1
[1,2,1,1,1,2] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,2,1,1,2,1] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,2,1,2,1,1] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,2,1,2,2] => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => ? = 1 + 1
[1,2,1,3,1] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
[1,2,2,1,1,1] => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,5),(3,4),(6,7),(8,9),(10,11),(12,13)]
 => ? = 3 + 1
[1,2,2,1,2] => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => ? = 1 + 1
[1,2,2,2,1] => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => ? = 1 + 1
[1,2,3,1,1] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,3,1,1,2] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,3,1,2,1] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 2 + 1
[1,3,1,3] => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[1,3,2,1,1] => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => ? = 4 + 1
[1,3,2,2] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => ? = 2 + 1
[1,3,3,1] => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => ? = 2 + 1
[1,4,1,1,1] => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,16),(8,9),(10,11),(12,13),(14,15)]
 => ? = 5 + 1
[2,2,2,2] => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => 2 = 1 + 1
[3,2,3] => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 3 = 2 + 1
[3,3,2] => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 3 = 2 + 1
[4,2,2] => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => 4 = 3 + 1

Description

The number of pairs with odd minimum in a perfect matching.

Matching statistic: St001553

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 13%

Values

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

Description

The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.

Matching statistic: St000329

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 13%

Values

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

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.