- FindStat

Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000022
(load all 10 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000241
(load all 6 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index

$i$ such that

$\pi_i = i+1$ considered cyclically.

Matching statistic: St000475
(load all 3 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000445
(load all 7 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St000215
(load all 3 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form

$(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic,

$\operatorname{adj_0}$ , counts adjacencies in the word with a zero appended.

$(\operatorname{adj_0}, \operatorname{des})$ and

$(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

Matching statistic: St000221
(load all 2 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong fixed points of a permutation.

$i$ is called a strong fixed point of

$\pi$ if 1.

$j < i$ implies

$\pi_j < \pi_i$ , and 2.

$j > i$ implies

$\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St000895
(load all 5 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ones on the main diagonal of an alternating sign matrix.

Matching statistic: St000986
(load all 2 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.

Matching statistic: St001008
(load all 4 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001017
(load all 2 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.

The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000674The number of hills of a Dyck path. St000894The trace of an alternating sign matrix. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001060The distinguishing index of a graph. St001903The number of fixed points of a parking function.