Processing math: 12%

Your data matches 218 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000708
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
Description
The product of the parts of an integer partition.
Matching statistic: St000657
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 15% ●values known / values provided: 74%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [3,3,3,1] => ? = 3
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [4,2,2,2] => ? = 4
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [4,2,2,2] => ? = 4
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [3,3,3,1] => ? = 3
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [4,2,2,2] => ? = 4
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [3,3,3,1] => ? = 3
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [3,3,2,2] => ? = 4
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [4,2,2,2] => ? = 4
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [3,3,3,1] => ? = 3
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => ? = 2
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
[[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [4,4,4,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14],[15]]
 => ? => ? = 8
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
[[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
[[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [5,4,3,2,1] => ? = 6
Description
The smallest part of an integer composition.
Matching statistic: St000017
(load all 3 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 15% ●values known / values provided: 74%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => ? = 3 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => ? = 3 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => ? = 3 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? = 4 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => ? = 3 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => ? = 2 - 1
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
[[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [4,4,4,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14],[15]]
 => ?
 => ? = 8 - 1
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
[[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
[[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
 => ? = 6 - 1
Description
The number of inversions of a standard tableau. Let T be a tableau. An inversion is an attacking pair (c,d) of the shape of T (see [[St000016]] for a definition of this) such that the entry of c in T is greater than the entry of d.
Matching statistic: St001803
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 15% ●values known / values provided: 74%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => ? = 3 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? = 4 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? = 4 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => ? = 3 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? = 4 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => ? = 3 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => ? = 4 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? = 4 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => ? = 3 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => ? = 2 - 1
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
[[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [4,4,4,2,1]
 => [5,4,3,3]
 => [[1,2,3,10,15],[4,5,6,14],[7,8,9],[11,12,13]]
 => ? = 8 - 1
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
[[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
[[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
 => [5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => ? = 6 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau T is the skew row-strict tableau obtained by gluing two copies of T such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals max, where \ell denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals 0, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000314
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 70%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,1,1,2,2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,3,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,3],[2,2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2],[3,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,3,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1],[2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2],[2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2],[2,3,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,3],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [10,7,8,9,4,5,6,1,2,3] => ? = 3
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,1,2,3,4] => ? = 4
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => ? = 4
Description
The number of left-to-right-maxima of a permutation. An integer \sigma_i in the one-line notation of a permutation \sigma is a '''left-to-right-maximum''' if there does not exist a j < i such that \sigma_j > \sigma_i. This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000654
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 70%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,1,1,2,2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,3,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,3],[2,2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2],[3,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,2],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,2],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2,3],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,3,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1],[2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2],[2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2],[2,3,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,2],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,3],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [10,7,8,9,4,5,6,1,2,3] => ? = 3
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,1,2,3,4] => ? = 4
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => ? = 4
Description
The first descent of a permutation. For a permutation \pi of \{1,\ldots,n\}, this is the smallest index 0 < i \leq n such that \pi(i) > \pi(i+1) where one considers \pi(n+1)=0.
Matching statistic: St001549
(load all 2 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 70%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[[1,1,1,2,2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,3,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,3,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3,3,3],[2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,3],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3,3],[2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,3,3,3],[2,2]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3,3],[2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,3],[2,2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3],[2,2,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2],[3,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3],[2,3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,2],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,2],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2,3],[2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,3,3],[2,2],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1],[2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2],[2,2,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2],[2,3,3],[3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,2],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,3],[2,2],[3,3]]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [10,7,8,9,4,5,6,1,2,3] => ? = 3 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 2 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,1,2,3,4] => ? = 4 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => ? = 4 - 1
Description
The number of restricted non-inversions between exceedances. This is for a permutation \sigma of length n given by \operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.
Matching statistic: St001520
(load all 4 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 67%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3,4,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3,4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3,5],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,4,5],[3]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,4,5],[2]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3],[4,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,4],[3,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,5],[3,4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,4],[2,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,5],[2,4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,4],[3],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,5],[3],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,4],[2],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,5],[2],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,4,5],[2],[3]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2],[3,4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2],[3,5],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3],[2,4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3],[2,5],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,4],[2,5],[3]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3],[2],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,4],[2],[3],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,5],[2],[3],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [4,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,1,4] => ? = 8
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
Description
The number of strict 3-descents. A '''strict 3-descent''' of a permutation \pi of \{1,2, \dots ,n \} is a pair (i,i+3) with i+3 \leq n and \pi(i) > \pi(i+3).
Matching statistic: St001556
(load all 5 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 67%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,4],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,4],[2]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3,4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2,4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3,4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,5],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,5],[2]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,5],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4,5],[2]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3,5],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4,5],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3,4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3,5],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,4,5],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[2,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3],[4,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,4],[3,5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2,3,4,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3,4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3,5],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,4,5],[3]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,4,5],[2]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3],[4,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,4],[3,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,5],[3,4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,4],[2,5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,5],[2,4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,4],[3],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2,5],[3],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,4],[2],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3,5],[2],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,4,5],[2],[3]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2],[3,4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2],[3,5],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3],[2,4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3],[2,5],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,4],[2,5],[3]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,3],[2],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,4],[2],[3],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,5],[2],[3],[4]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => ? = 4
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => ? = 4
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => ? = 3
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [4,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,1,4] => ? = 8
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ? = 6
Description
The number of inversions of the third entry of a permutation. This is, for a permutation \pi of length n, \# \{3 < k \leq n \mid \pi(3) > \pi(k)\}. The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001526
(load all 4 compositions to match this statistic)
Mapping
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 61%●distinct values known / distinct values provided: 15%
Values
[[1,2,3,4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,4],[3]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,4],[2]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[3,4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3],[2,4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3],[2],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4],[2],[3]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,4,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,4,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,3,4,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,5],[3]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,5],[2]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,5],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4,5],[2]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,5],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4,5],[3]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,3,4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,3,5],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,4,5],[3]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[3,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3],[2,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[4,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4],[2,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3],[4,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4],[3,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,3],[4,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,4],[3,5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3],[2],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,5],[2],[3]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4],[2],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,5],[2],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4],[3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,5],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,4],[3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2,5],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,1,1,2,3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,2,3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,3,3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,4,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,2,3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,2,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,3,4],[2]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,2,3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,2,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,3,4],[2]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,3,3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,3,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,3,3,4],[2]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,4],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,4,4,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,4,4,4],[2]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,2],[3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,3],[2,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,4],[2,3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,2],[3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,3],[2,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,4],[2,3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,3],[3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4],[3,3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,3,3],[2,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4],[4,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,4,4],[3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,4,4],[2,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,3],[2],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1,4],[2],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,3],[2],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2,4],[2],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,3],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,3,3],[2],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,3,4],[2],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,4,4],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,4,4],[2],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,4,4,4],[2],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1],[2,3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2],[2,3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3],[3,3,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3],[4,4,4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1],[2,3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,1,1],[2,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2],[2,3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,2],[2,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3],[3,3],[4]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,3,3],[2,4],[3]]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 2
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
The following 208 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000090The variation of a composition. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001964The interval resolution global dimension of a poset. 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. St001260The permanent of an alternating sign matrix. 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). St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra A/I when I is the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001256Number of simple reflexive modules that are 2-stable reflexive. St001344The neighbouring number of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000177The number of free tiles in the pattern. St000178Number of free entries. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000666The number of right tethers of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001684The reduced word complexity of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001868The number of alignments of type NE of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000735The last entry on the main diagonal of a standard tableau. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000135The number of lucky cars of the parking function. St001927Sparre Andersen's number of positives of a signed permutation. St000540The sum of the entries of a parking function minus its length. St000165The sum of the entries of a parking function. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001768The number of reduced words of a signed permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. 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). St001926Sparre Andersen's position of the maximum of a signed permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000072The number of circled entries. St000073The number of boxed entries. St000077The number of boxed and circled entries. St001721The degree of a binary word. St001171The vector space dimension of Ext_A^1(I_o,A) when I_o is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(x^n). St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St000016The number of attacking pairs of a standard tableau. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St000068The number of minimal elements in a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000907The number of maximal antichains of minimal length in a poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000170The trace of a semistandard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. 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. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000080The rank of the poset. St000189The number of elements in the poset. St001624The breadth of a lattice. St000635The number of strictly order preserving maps of a poset into itself. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000043The number of crossings plus two-nestings of a perfect matching. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001071The beta invariant of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001354The number of series nodes in the modular decomposition of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001754The number of tolerances of a finite lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000529The number of permutations whose descent word is the given binary word. St000633The size of the automorphism group of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001644The dimension of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000081The number of edges of a graph. St000454The largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000911The number of maximal antichains of maximal size in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules S with dim Ext^1(S,A)=1 in the incidence algebra A of the poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001734The lettericity of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001410The minimal entry of a semistandard tableau. St001409The maximal entry of a semistandard tableau.