- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000053

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: 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]
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0

Description

The number of valleys of the Dyck path.

Matching statistic: St001068

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[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]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[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]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[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]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[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]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St000331

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
 => [2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
 => [3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2
[7,1,1] => [[7,7,7],[6,6]]
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[8,1] => [[8,8],[7]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6

Description

The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''

$A^{k}$

$B^{k}$ ''' for any '''

${k ≥ 1}$ ''', where '''

${A}$ ''' is a down-step and '''

${B}$ ''' is a up-step.

Matching statistic: St001169

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
 => [2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
 => [3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2
[7,1,1] => [[7,7,7],[6,6]]
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[8,1] => [[8,8],[7]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6

Description

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St001509

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
 => [2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
 => [3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2
[7,1,1] => [[7,7,7],[6,6]]
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[8,1] => [[8,8],[7]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6

Description

The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. Given two lattice paths

$U,L$ from

$(0,0)$ to

$(d,n-d)$ , [1] describes a bijection between lattice paths weakly between

$U$ and

$L$ and subsets of

$\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid

$M[U,L]$ . This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly below the diagonal and relative to the trivial lower boundary.

Matching statistic: St000015

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%

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]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[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]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[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]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[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]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 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]
 => [1,1,0,0]
 => 1 = 0 + 1
[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]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 1 = 0 + 1
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 1
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
 => [2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
 => [3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
[7,1,1] => [[7,7,7],[6,6]]
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4 + 1
[8,1] => [[8,8],[7]]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1

Description

The number of peaks of a Dyck path.