Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000784
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Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 1
[[1,2]]
 => [2]
 => 2
[[1],[2]]
 => [1,1]
 => 2
[[1,2,3]]
 => [3]
 => 3
[[1,3],[2]]
 => [2,1]
 => 2
[[1,2],[3]]
 => [2,1]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 3
[[1,2,3,4]]
 => [4]
 => 4
[[1,3,4],[2]]
 => [3,1]
 => 3
[[1,2,4],[3]]
 => [3,1]
 => 3
[[1,2,3],[4]]
 => [3,1]
 => 3
[[1,3],[2,4]]
 => [2,2]
 => 2
[[1,2],[3,4]]
 => [2,2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => 3
[[1,3],[2],[4]]
 => [2,1,1]
 => 3
[[1,2],[3],[4]]
 => [2,1,1]
 => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => 4
[[1,2,3,4,5]]
 => [5]
 => 5
[[1,3,4,5],[2]]
 => [4,1]
 => 4
[[1,2,4,5],[3]]
 => [4,1]
 => 4
[[1,2,3,5],[4]]
 => [4,1]
 => 4
[[1,2,3,4],[5]]
 => [4,1]
 => 4
[[1,3,5],[2,4]]
 => [3,2]
 => 3
[[1,2,5],[3,4]]
 => [3,2]
 => 3
[[1,3,4],[2,5]]
 => [3,2]
 => 3
[[1,2,4],[3,5]]
 => [3,2]
 => 3
[[1,2,3],[4,5]]
 => [3,2]
 => 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 3
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 3
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 3
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 3
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 3
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 3
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 3
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 3
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 3
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 3
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 3
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => 4
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => 4
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => 4
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => 4
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => 5
[[1,2,3,4,5,6]]
 => [6]
 => 6
[[1,3,4,5,6],[2]]
 => [5,1]
 => 5
[[1,2,4,5,6],[3]]
 => [5,1]
 => 5
[[1,2,3,5,6],[4]]
 => [5,1]
 => 5
[[1,2,3,4,6],[5]]
 => [5,1]
 => 5
[[1,2,3,4,5],[6]]
 => [5,1]
 => 5
[[1,3,5,6],[2,4]]
 => [4,2]
 => 4
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St000956
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 61%●distinct values known / distinct values provided: 42%
Values
[[1]]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[[1,2]]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[[1],[2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[[1,2,3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[[1,3],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[[1,2],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[[1],[2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[[1,2,3,4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
[[1,3,4],[2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[[1,2,4],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[[1,2,3],[4]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[[1,3],[2,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[[1,2],[3,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 3
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 3
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 5
[[1,3,4,5],[2]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 4
[[1,2,4,5],[3]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 4
[[1,2,3,5],[4]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 4
[[1,2,3,4],[5]]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 4
[[1,3,5],[2,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[[1,2,5],[3,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[[1,3,4],[2,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[[1,2,4],[3,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[[1,2,3],[4,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 4
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 4
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 4
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 4
[[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] => 5
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 5
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 5
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 5
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 5
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 5
[[1,3,5,6],[2,4]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4
[[1,2,5,6],[3,4]]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 6
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ? = 7
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 6
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 6
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 6
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 6
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 6
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 6
[[1,7],[2],[3],[4],[5],[6]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 6
[[1,6],[2],[3],[4],[5],[7]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 6
[[1,5],[2],[3],[4],[6],[7]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 6
[[1,4],[2],[3],[5],[6],[7]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 6
[[1,3],[2],[4],[5],[6],[7]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 6
[[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 6
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ? = 7
[[1,2,3,4,5,6,7,8]]
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ? = 8
[[1,3,4,5,6,7,8],[2]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,2,4,5,6,7,8],[3]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,2,3,5,6,7,8],[4]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,2,3,4,6,7,8],[5]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,2,3,4,5,7,8],[6]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,2,3,4,5,6,8],[7]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,2,3,4,5,6,7],[8]]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ? = 7
[[1,3,5,6,7,8],[2,4]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,5,6,7,8],[3,4]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,3,4,6,7,8],[2,5]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,4,6,7,8],[3,5]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,6,7,8],[4,5]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,3,4,5,7,8],[2,6]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,4,5,7,8],[3,6]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,5,7,8],[4,6]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,4,7,8],[5,6]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,3,4,5,6,8],[2,7]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,4,5,6,8],[3,7]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,5,6,8],[4,7]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,4,6,8],[5,7]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,4,5,8],[6,7]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,3,4,5,6,7],[2,8]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,4,5,6,7],[3,8]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,5,6,7],[4,8]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,4,6,7],[5,8]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,4,5,7],[6,8]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,2,3,4,5,6],[7,8]]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 6
[[1,4,5,6,7,8],[2],[3]]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 6
[[1,3,5,6,7,8],[2],[4]]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 6
[[1,2,5,6,7,8],[3],[4]]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 6
[[1,3,4,6,7,8],[2],[5]]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 6
[[1,2,4,6,7,8],[3],[5]]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 6
[[1,2,3,6,7,8],[4],[5]]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 6
Description
The maximal displacement of a permutation. This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$. This statistic without the absolute value is the maximal drop size [[St000141]].