Your data matches 314 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001252
Mapping
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 2
[3,1]
 => 0
[2,2]
 => 2
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 2
[3,2]
 => 1
[3,1,1]
 => 0
[2,2,1]
 => 2
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 3
[5,1]
 => 0
[4,2]
 => 3
[4,1,1]
 => 2
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,2]
 => 3
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
Description
Half the sum of the even parts of a partition.
Matching statistic: St000143
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => 3
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 3
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => 0
Description
The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000228
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => []
 => 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => [2]
 => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => []
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => [3]
 => 3
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => [3]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => []
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => []
 => 0
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000459
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => []
 => 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => [2]
 => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => []
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => [3]
 => 3
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => [3]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => []
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => []
 => 0
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000589
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,5,6},{3},{4}}
 => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => {{1,2,6},{3,4},{5}}
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,6,7},{3},{4},{5}}
 => 3
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2},{3},{4}}
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => {{1,2,7},{3,4},{5},{6}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St001801
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => [1] => 0
[2]
 => [[1,2]]
 => [1,2] => [2,1] => 1
[1,1]
 => [[1],[2]]
 => [2,1] => [1,2] => 0
[3]
 => [[1,2,3]]
 => [1,2,3] => [3,2,1] => 0
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 2
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [1,4,3,2] => 0
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => 0
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,5,4,3,2] => 2
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,5,4,3] => 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 0
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,2,5,4] => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,5,4] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 3
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => 3
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [1,2,6,5,4,3] => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 0
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [1,3,2,6,5,4] => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [1,2,3,6,5,4] => 0
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [1,2,4,3,6,5] => 2
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
Description
Half the number of preimage-image pairs of different parity in a permutation.
Matching statistic: St000355
(load all 9 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,3,4,5,2] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [2,3,4,5,1,6] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [3,4,2,1,5] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,4,5,3,2] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [1,3,4,5,6,2] => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => ? = 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,3,4,2,1,6] => [2,4,5,3,1,6] => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => 3
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,2,4,1,5] => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,3,1,5,4] => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,3,5,2] => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,6,4,5,3,2] => [1,3,5,6,4,2] => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => 0
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St001377
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,3,4,5,2] => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [2,3,4,5,1,6] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [3,4,2,1,5] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,4,5,3,2] => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [1,3,4,5,6,2] => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => ? = 0
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,3,4,2,1,6] => [2,4,5,3,1,6] => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => 0
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,2,4,1,5] => 0
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,3,1,5,4] => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,3,5,2] => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,6,4,5,3,2] => [1,3,5,6,4,2] => 3
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => 1
Description
The major index minus the number of inversions of a permutation. This is, the difference between [[St000004]] and [[St000018]].
Matching statistic: St000359
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [2,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => [1,4,3,2,6,5] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => [2,4,3,6,5,1] => 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,.],.]]
 => [1,4,3,2,6,7,5] => ? ∊ {0,3}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => 0
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => 0
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => [2,3,4,6,5,1] => 3
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => [2,4,3,6,7,5,1] => ? ∊ {0,3}
Description
The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St001229
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001229: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? ∊ {1,3}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? ∊ {1,3}
Description
The vector space dimension of the first extension group between the Jacobson radical J and J^2. The vector space dimension of $Ext_A^1(J,J^2)$.
The following 304 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000005The bounce statistic of a Dyck path. St000516The number of stretching pairs of a permutation. St001161The major index north count of a Dyck path. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000478Another weight of a partition according to Alladi. St000693The modular (standard) major index of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000248The number of anti-singletons of a set partition. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000498The lcs statistic of a set partition. St000503The maximal difference between two elements in a common block. St000538The number of even inversions of a permutation. St000674The number of hills of a Dyck path. St000836The number of descents of distance 2 of a permutation. St000947The major index east count of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001721The degree of a binary word. St000454The largest eigenvalue of a graph if it is integral. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001130The number of two successive successions in a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001176The size of a partition minus its first part. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000006The dinv of a Dyck path. St000010The length of the partition. St000012The area of a Dyck path. St000053The number of valleys of the Dyck path. St000148The number of odd parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000160The multiplicity of the smallest part of a partition. St000185The weighted size of a partition. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000644The number of graphs with given frequency partition. St000784The maximum of the length and the largest part of the integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000867The sum of the hook lengths in the first row of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001127The sum of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001435The number of missing boxes in the first row. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000761The number of ascents in an integer composition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001556The number of inversions of the third entry of a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001948The number of augmented double ascents of a permutation. St000765The number of weak records in an integer composition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001487The number of inner corners of a skew partition. St001737The number of descents of type 2 in a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000638The number of up-down runs of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001541The Gini index of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001488The number of corners of a skew partition. St000259The diameter of a connected graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000383The last part of an integer composition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000456The monochromatic index of a connected graph. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000260The radius of a connected graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000932The number of occurrences of the pattern UDU in a Dyck path. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001098The 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 vertex labelled trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000455The second largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000649The number of 3-excedences of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000939The number of characters of the symmetric group whose value on the partition is positive. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000492The rob statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000942The number of critical left to right maxima of the parking functions. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000542The number of left-to-right-minima of a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000993The multiplicity of the largest part of an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001568The smallest positive integer that does not appear twice in the partition. St001651The Frankl number of a lattice. St000356The number of occurrences of the pattern 13-2. St000834The number of right outer peaks of a permutation. St000007The number of saliances of the permutation. St000091The descent variation of a composition. St000365The number of double ascents of a permutation. St000562The number of internal points of a set partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000565The major index of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000779The tier of a permutation. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001060The distinguishing index of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000717The number of ordinal summands of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral.