- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000503
(load all 9 compositions to match this statistic)

Mapping

St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal difference between two elements in a common block.

Matching statistic: St000442

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000730

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal arc length of a set partition. The arcs of a set partition are those

$i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is

$0$ .

Matching statistic: St000013

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000209
(load all 9 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 86%

Values

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

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 9 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 86%

Values

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

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: St000141
(load all 5 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [2,1] => 1
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 3
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => 3
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => 3
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => 4
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => 4
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,1,3,2,4] => 4
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,5,2,3] => 3
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => 3
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,1,4,3,2] => 4
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,1,3,4,2] => 4
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,1,3,4] => 4
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,5,1,3,2] => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,1,3,5] => 3
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,1,2,3] => 4
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,5,1,2,4] => 3
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2,5] => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,1,4,3] => 4
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => 3
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,1,4] => 4
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,2,1,3] => 3
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => 3
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 6
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 6
{{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ? = 5
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 6
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => ? = 3
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => ? = 3
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 6
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => ? = 3
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => ? = 6
{{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => [6,1,2,7,3,5,4] => ? = 5
{{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => [6,1,2,4,3,5,7] => ? = 5
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 6
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [5,1,2,7,3,4,6] => ? = 4
{{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => ? = 6
{{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => [5,1,2,7,3,6,4] => ? = 4
{{1,2,3,5},{4},{6,7}}
 => [2,3,5,4,1,7,6] => [5,1,2,4,3,7,6] => ? = 4
{{1,2,3,5},{4},{6},{7}}
 => [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => ? = 4
{{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => ? = 6
{{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => [6,1,2,7,4,3,5] => ? = 5
{{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => [6,1,2,5,4,3,7] => ? = 5
{{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => ? = 6
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => ? = 3
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => ? = 2
{{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => ? = 6
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => [3,1,2,7,4,6,5] => ? = 3
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => ? = 2
{{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => ? = 2
{{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => ? = 6
{{1,2,3,6},{4,7},{5}}
 => [2,3,6,7,5,1,4] => [6,1,2,7,5,3,4] => ? = 5
{{1,2,3,6},{4},{5,7}}
 => [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => ? = 5
{{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => [6,1,2,4,5,3,7] => ? = 5
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => ? = 6
{{1,2,3},{4,6,7},{5}}
 => [2,3,1,6,5,7,4] => [3,1,2,7,5,4,6] => ? = 3
{{1,2,3},{4,6},{5,7}}
 => [2,3,1,6,7,4,5] => [3,1,2,6,7,4,5] => ? = 2
{{1,2,3},{4,6},{5},{7}}
 => [2,3,1,6,5,4,7] => [3,1,2,6,5,4,7] => ? = 2
{{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => ? = 6
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => [3,1,2,7,6,5,4] => ? = 3
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => ? = 2
{{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => ? = 2
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => ? = 6
{{1,2,3},{4,7},{5},{6}}
 => [2,3,1,7,5,6,4] => [3,1,2,7,5,6,4] => ? = 3
{{1,2,3},{4},{5,7},{6}}
 => [2,3,1,4,7,6,5] => [3,1,2,4,7,6,5] => ? = 2
{{1,2,4,5,6,7},{3}}
 => [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => ? = 6
{{1,2,4,5,6},{3,7}}
 => [2,4,7,5,6,1,3] => [6,1,7,2,4,5,3] => ? = 5
{{1,2,4,5,6},{3},{7}}
 => [2,4,3,5,6,1,7] => [6,1,3,2,4,5,7] => ? = 5
{{1,2,4,5,7},{3,6}}
 => [2,4,6,5,7,3,1] => [7,1,6,2,4,3,5] => ? = 6
{{1,2,4,5},{3,6,7}}
 => [2,4,6,5,1,7,3] => [5,1,7,2,4,3,6] => ? = 4
{{1,2,4,5},{3,6},{7}}
 => [2,4,6,5,1,3,7] => [5,1,6,2,4,3,7] => ? = 4
{{1,2,4,5,7},{3},{6}}
 => [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => ? = 6
{{1,2,4,5},{3,7},{6}}
 => [2,4,7,5,1,6,3] => [5,1,7,2,4,6,3] => ? = 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: St000651
(load all 3 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St001207
(load all 7 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 57%

Values

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

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