Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001176
(load all 6 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 0
[1,2] => [1,1]
 => 1
[2,1] => [2]
 => 0
[1,2,3] => [1,1,1]
 => 2
[1,3,2] => [2,1]
 => 1
[2,1,3] => [2,1]
 => 1
[2,3,1] => [2,1]
 => 1
[3,1,2] => [2,1]
 => 1
[3,2,1] => [3]
 => 0
[1,2,3,4] => [1,1,1,1]
 => 3
[1,2,4,3] => [2,1,1]
 => 2
[1,3,2,4] => [2,1,1]
 => 2
[1,3,4,2] => [2,1,1]
 => 2
[1,4,2,3] => [2,1,1]
 => 2
[1,4,3,2] => [3,1]
 => 1
[2,1,3,4] => [2,1,1]
 => 2
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [2,1,1]
 => 2
[2,3,4,1] => [2,1,1]
 => 2
[2,4,1,3] => [2,1,1]
 => 2
[2,4,3,1] => [3,1]
 => 1
[3,1,2,4] => [2,1,1]
 => 2
[3,1,4,2] => [2,2]
 => 2
[3,2,1,4] => [3,1]
 => 1
[3,2,4,1] => [3,1]
 => 1
[3,4,1,2] => [2,1,1]
 => 2
[3,4,2,1] => [3,1]
 => 1
[4,1,2,3] => [2,1,1]
 => 2
[4,1,3,2] => [3,1]
 => 1
[4,2,1,3] => [3,1]
 => 1
[4,2,3,1] => [3,1]
 => 1
[4,3,1,2] => [3,1]
 => 1
[4,3,2,1] => [4]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => 4
[1,2,3,5,4] => [2,1,1,1]
 => 3
[1,2,4,3,5] => [2,1,1,1]
 => 3
[1,2,4,5,3] => [2,1,1,1]
 => 3
[1,2,5,3,4] => [2,1,1,1]
 => 3
[1,2,5,4,3] => [3,1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => 3
[1,3,2,5,4] => [2,2,1]
 => 3
[1,3,4,2,5] => [2,1,1,1]
 => 3
[1,3,4,5,2] => [2,1,1,1]
 => 3
[1,3,5,2,4] => [2,1,1,1]
 => 3
[1,3,5,4,2] => [3,1,1]
 => 2
[1,4,2,3,5] => [2,1,1,1]
 => 3
[1,4,2,5,3] => [2,2,1]
 => 3
[1,4,3,2,5] => [3,1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => 2
[1,4,5,2,3] => [2,1,1,1]
 => 3
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000211
(load all 36 compositions to match this statistic)
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
 => {{1}}
 => 0
[1,2] => [[1,2]]
 => {{1,2}}
 => 1
[2,1] => [[1],[2]]
 => {{1},{2}}
 => 0
[1,2,3] => [[1,2,3]]
 => {{1,2,3}}
 => 2
[1,3,2] => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[2,1,3] => [[1,3],[2]]
 => {{1,3},{2}}
 => 1
[2,3,1] => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[3,1,2] => [[1,3],[2]]
 => {{1,3},{2}}
 => 1
[3,2,1] => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0
[1,2,3,4] => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[1,2,4,3] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[1,3,2,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 2
[1,3,4,2] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[1,4,2,3] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 2
[1,4,3,2] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[2,1,3,4] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2
[2,1,4,3] => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2
[2,3,1,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 2
[2,3,4,1] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[2,4,1,3] => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
[2,4,3,1] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[3,1,2,4] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2
[3,1,4,2] => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2
[3,2,1,4] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1
[3,2,4,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1
[3,4,1,2] => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
[3,4,2,1] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[4,1,2,3] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2
[4,1,3,2] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1
[4,2,1,3] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1
[4,2,3,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1
[4,3,1,2] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1
[4,3,2,1] => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0
[1,2,3,4,5] => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000228
(load all 8 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => []
 => 0
[1,2] => [1,1]
 => [1]
 => 1
[2,1] => [2]
 => []
 => 0
[1,2,3] => [1,1,1]
 => [1,1]
 => 2
[1,3,2] => [2,1]
 => [1]
 => 1
[2,1,3] => [2,1]
 => [1]
 => 1
[2,3,1] => [2,1]
 => [1]
 => 1
[3,1,2] => [2,1]
 => [1]
 => 1
[3,2,1] => [3]
 => []
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 3
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 2
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 2
[1,3,4,2] => [2,1,1]
 => [1,1]
 => 2
[1,4,2,3] => [2,1,1]
 => [1,1]
 => 2
[1,4,3,2] => [3,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 2
[2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4] => [2,1,1]
 => [1,1]
 => 2
[2,3,4,1] => [2,1,1]
 => [1,1]
 => 2
[2,4,1,3] => [2,1,1]
 => [1,1]
 => 2
[2,4,3,1] => [3,1]
 => [1]
 => 1
[3,1,2,4] => [2,1,1]
 => [1,1]
 => 2
[3,1,4,2] => [2,2]
 => [2]
 => 2
[3,2,1,4] => [3,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => 1
[3,4,1,2] => [2,1,1]
 => [1,1]
 => 2
[3,4,2,1] => [3,1]
 => [1]
 => 1
[4,1,2,3] => [2,1,1]
 => [1,1]
 => 2
[4,1,3,2] => [3,1]
 => [1]
 => 1
[4,2,1,3] => [3,1]
 => [1]
 => 1
[4,2,3,1] => [3,1]
 => [1]
 => 1
[4,3,1,2] => [3,1]
 => [1]
 => 1
[4,3,2,1] => [4]
 => []
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 3
[1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 2
[1,4,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => 3
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => 3
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 2
[1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => 0
[1,2] => [2]
 => [1,1]
 => 1
[2,1] => [1,1]
 => [2]
 => 0
[1,2,3] => [3]
 => [1,1,1]
 => 2
[1,3,2] => [2,1]
 => [3]
 => 1
[2,1,3] => [2,1]
 => [3]
 => 1
[2,3,1] => [2,1]
 => [3]
 => 1
[3,1,2] => [2,1]
 => [3]
 => 1
[3,2,1] => [1,1,1]
 => [2,1]
 => 0
[1,2,3,4] => [4]
 => [1,1,1,1]
 => 3
[1,2,4,3] => [3,1]
 => [2,1,1]
 => 2
[1,3,2,4] => [3,1]
 => [2,1,1]
 => 2
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 2
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 2
[1,4,3,2] => [2,1,1]
 => [2,2]
 => 1
[2,1,3,4] => [3,1]
 => [2,1,1]
 => 2
[2,1,4,3] => [2,2]
 => [4]
 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 2
[2,3,4,1] => [3,1]
 => [2,1,1]
 => 2
[2,4,1,3] => [2,2]
 => [4]
 => 2
[2,4,3,1] => [2,1,1]
 => [2,2]
 => 1
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 2
[3,1,4,2] => [2,2]
 => [4]
 => 2
[3,2,1,4] => [2,1,1]
 => [2,2]
 => 1
[3,2,4,1] => [2,1,1]
 => [2,2]
 => 1
[3,4,1,2] => [2,2]
 => [4]
 => 2
[3,4,2,1] => [2,1,1]
 => [2,2]
 => 1
[4,1,2,3] => [3,1]
 => [2,1,1]
 => 2
[4,1,3,2] => [2,1,1]
 => [2,2]
 => 1
[4,2,1,3] => [2,1,1]
 => [2,2]
 => 1
[4,2,3,1] => [2,1,1]
 => [2,2]
 => 1
[4,3,1,2] => [2,1,1]
 => [2,2]
 => 1
[4,3,2,1] => [1,1,1,1]
 => [3,1]
 => 0
[1,2,3,4,5] => [5]
 => [1,1,1,1,1]
 => 4
[1,2,3,5,4] => [4,1]
 => [2,1,1,1]
 => 3
[1,2,4,3,5] => [4,1]
 => [2,1,1,1]
 => 3
[1,2,4,5,3] => [4,1]
 => [2,1,1,1]
 => 3
[1,2,5,3,4] => [4,1]
 => [2,1,1,1]
 => 3
[1,2,5,4,3] => [3,1,1]
 => [4,1]
 => 2
[1,3,2,4,5] => [4,1]
 => [2,1,1,1]
 => 3
[1,3,2,5,4] => [3,2]
 => [5]
 => 3
[1,3,4,2,5] => [4,1]
 => [2,1,1,1]
 => 3
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 3
[1,3,5,2,4] => [3,2]
 => [5]
 => 3
[1,3,5,4,2] => [3,1,1]
 => [4,1]
 => 2
[1,4,2,3,5] => [4,1]
 => [2,1,1,1]
 => 3
[1,4,2,5,3] => [3,2]
 => [5]
 => 3
[1,4,3,2,5] => [3,1,1]
 => [4,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [4,1]
 => 2
[1,4,5,2,3] => [3,2]
 => [5]
 => 3
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000507
(load all 4 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ 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]
 => [[1,2]]
 => 2 = 1 + 1
[2,1] => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[1,2,3] => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[1,2,4,3] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,3,2,4] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[2,1,3,4] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,3,4,1] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[2,4,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[3,2,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,4,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[4,1,2,3] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[4,1,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[4,2,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[4,3,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[1,2,3,5,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,2,4,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,2,4,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,2,5,3,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,3,2,4,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,3,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,3,4,2,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,4,2,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,4,5,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000738
(load all 3 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[2,1] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[2,3,1] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[3,1,2] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[3,2,1] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,3,4,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,4,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,4,3,2] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[2,3,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[2,3,4,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[2,4,1,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[3,1,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,2,1,4] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[3,4,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[3,4,2,1] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[4,1,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[4,2,3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[4,3,1,2] => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[4,3,2,1] => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,2,4,5,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,2,5,3,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,3,4,5,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,3,5,2,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[1,4,2,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,4,2,5,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[1,4,5,2,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000074
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [[1]]
 => 0
[1,2] => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 1
[2,1] => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 0
[1,2,3] => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 0
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,2,4,3] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,3,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[2,1,3,4] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,3,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[2,4,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[3,2,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[3,4,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[4,1,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[4,1,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[4,2,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[4,3,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
[1,2,3,5,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,2,4,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,2,4,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,2,5,3,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,3,2,4,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,3,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,3,4,2,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,4,2,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[1,4,5,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
Description
The number of special entries. An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000141
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 0
[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[2,3,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[3,1,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[3,2,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 3
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[1,3,4,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[1,4,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[1,4,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,3,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[2,3,4,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[2,4,1,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[3,1,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[3,2,1,4] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[3,4,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[3,4,2,1] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[4,1,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[4,2,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[4,3,1,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[4,3,2,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 4
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,2,4,5,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,2,5,3,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,2,5,4,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 3
[1,3,4,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,3,4,5,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,3,5,2,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
[1,4,2,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,4,2,5,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 3
[1,4,3,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
[1,4,5,2,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
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: St000157
(load all 4 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [[1]]
 => 0
[1,2] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[2,1] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
[1,2,3] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
[1,2,4,3] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[1,3,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[2,1,3,4] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[2,3,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,4,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[3,2,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[4,1,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[4,1,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[4,2,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[4,3,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
[1,2,3,5,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,2,4,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,2,4,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,2,5,3,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
[1,3,2,4,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,3,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
[1,3,4,2,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
[1,4,2,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
[1,4,5,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000245
(load all 16 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 0
[1,2] => [2]
 => [[1,2]]
 => [1,2] => 1
[2,1] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3] => [3]
 => [[1,2,3]]
 => [1,2,3] => 2
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
[1,2,4,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,3,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[2,1,3,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[2,3,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,4,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[3,2,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[3,4,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,1,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[4,1,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,2,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,3,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
[1,2,3,5,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,2,4,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,2,4,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,2,5,3,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 2
[1,3,2,4,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,3,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 3
[1,3,4,2,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 3
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 2
[1,4,2,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 3
[1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 3
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 2
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 2
[1,4,5,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 3
Description
The number of ascents of a permutation.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St000010The length of the partition. St000054The first entry of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000839The largest opener of a set partition. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St001726The number of visible inversions of a permutation. St000459The hook length of the base cell of a partition. 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. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000369The dinv deficit of a Dyck path. St000362The size of a minimal vertex cover of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001298The number of repeated entries in the Lehmer code of a permutation. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000209Maximum difference of elements in cycles. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001626The number of maximal proper sublattices of a lattice.