- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000503
(load all 5 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal difference between two elements in a common block.

Matching statistic: St000141

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000209
(load all 3 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 75%

Values

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

Description

Maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the maximum of this value over all cycles in the permutation.

Matching statistic: St000956
(load all 3 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 75%

Values

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

Description

The maximal displacement of a permutation. This is

$\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation

$\pi$ of

$\{1,\ldots,n\}$ . This statistic without the absolute value is the maximal drop size [[St000141]].

Matching statistic: St001879
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 62%

Values

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

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001880
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 62%

Values

[1,2] => [1,0,1,0]
 => [2,1] => ([],2)
 => ? = 0 + 1
[2,1] => [1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ? = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ? = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ? = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ? = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ? = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ? = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ? = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ? = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ? = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ? = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ? = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ? = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ? = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ? = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ? = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ? = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ? = 2 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ? = 2 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 3 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 3 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 3 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 3 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 3 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 3 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ? = 1 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St001232

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 62%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001207
(load all 5 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%

Values

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

Description

The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .