- FindStat

Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000784
(load all 2 compositions to match this statistic)

Mapping

St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1
[2]
 => 2
[1,1]
 => 2
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 3
[4]
 => 4
[3,1]
 => 3
[2,2]
 => 2
[2,1,1]
 => 3
[1,1,1,1]
 => 4
[5]
 => 5
[4,1]
 => 4
[3,2]
 => 3
[3,1,1]
 => 3
[2,2,1]
 => 3
[2,1,1,1]
 => 4
[1,1,1,1,1]
 => 5
[6]
 => 6
[5,1]
 => 5
[4,2]
 => 4
[4,1,1]
 => 4
[3,3]
 => 3
[3,2,1]
 => 3
[3,1,1,1]
 => 4
[2,2,2]
 => 3
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 5
[1,1,1,1,1,1]
 => 6
[7]
 => 7
[6,1]
 => 6
[5,2]
 => 5
[5,1,1]
 => 5
[4,3]
 => 4
[4,2,1]
 => 4
[4,1,1,1]
 => 4
[3,3,1]
 => 3
[3,2,2]
 => 3
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 5
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 6
[1,1,1,1,1,1,1]
 => 7
[8]
 => 8
[7,1]
 => 7
[6,2]
 => 6
[6,1,1]
 => 6
[5,3]
 => 5
[5,2,1]
 => 5

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: St000727

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%

Values

[1]
 => [1,0,1,0]
 => [2,1] => [1,2] => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => 3 = 2 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 3 = 2 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => 4 = 3 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,4,5] => 5 = 4 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => 4 = 3 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => 4 = 3 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 5 = 4 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => 6 = 5 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [1,3,4,2,5] => 5 = 4 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => 4 = 3 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => 4 = 3 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => 4 = 3 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => 5 = 4 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6 = 5 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [1,3,4,5,2,6] => 6 = 5 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,2,4,3,5] => 5 = 4 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => 5 = 4 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,3,4,5] => 5 = 4 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => 4 = 3 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => 5 = 4 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => 4 = 3 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [1,2,4,5,6,3] => 6 = 5 + 1
[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] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[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] => [1,2,3,4,5,6,7,8] => ? ∊ {7,7} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => 7 = 6 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [1,2,4,5,3,6] => 6 = 5 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [1,4,5,2,3,6] => 6 = 5 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,3,4,5] => 5 = 4 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => 5 = 4 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => 5 = 4 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => 5 = 4 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => 4 = 3 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [1,2,3,5,6,4] => 6 = 5 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => 5 = 4 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [1,2,5,6,3,4] => 6 = 5 + 1
[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] => [1,2,4,5,6,7,3] => 7 = 6 + 1
[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] => [1,2,3,4,5,6,7,8] => ? ∊ {7,7} + 1
[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] => [1,2,3,4,5,6,7,8,9] => ? ∊ {7,7,8,8} + 1
[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] => [1,3,4,5,6,7,2,8] => ? ∊ {7,7,8,8} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,2,4,5,6,3,7] => 7 = 6 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [1,4,5,6,2,3,7] => 7 = 6 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,2,3,5,4,6] => 6 = 5 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [1,4,5,2,3,6] => 6 = 5 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [1,5,2,3,4,6] => 6 = 5 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [1,2,3,4,5,6] => 6 = 5 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => 5 = 4 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => 5 = 4 + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => [1,2,4,5,6,7,8,3] => ? ∊ {7,7,8,8} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => ? ∊ {7,7,8,8} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2,9] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => [1,2,4,5,6,7,3,8] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => [1,4,5,6,7,2,3,8] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,1,5,6] => [1,5,6,2,3,4,7] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => [1,2,3,5,6,7,8,4] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => [1,2,4,5,6,7,8,9,3] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {6,7,7,7,7,8,8,9,9} + 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10,11] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2,10] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => [1,2,4,5,6,7,8,3,9] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3,9] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => [1,2,3,5,6,7,4,8] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => [1,4,5,6,7,2,3,8] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => [1,5,6,7,2,3,4,8] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,1,5,6] => [1,5,6,2,4,3,7] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => [1,2,3,4,6,7,8,5] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => [1,2,3,5,6,7,8,9,4] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => [1,2,5,6,7,8,9,3,4] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => [1,2,4,5,6,7,8,9,10,3] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,10,11] => ? ∊ {6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10} + 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,1,3,4,5,6,7,8,9,10] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,1,2,4,5,6,7,8,9] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,3,1,4,5,6,7,8,9] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [9,4,1,2,3,5,6,7,8] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,4,1,5,6,7,8] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,1,3,5,6,7] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,3,1,5,6] => [1,5,6,2,3,4,7] => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [7,3,4,2,1,5,6] => [1,5,6,2,3,4,7] => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ? => ? ∊ {6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} + 1

Description

The largest label of a leaf in the binary search tree associated with the permutation. Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.

Matching statistic: St000141

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 83%

Values

[1]
 => [1,0,1,0]
 => [2,1] => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1,3] => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,3,1] => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => 3 = 4 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 4 = 5 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,5,4,3,1] => 3 = 4 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => 3 = 4 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [5,4,3,2,1,6] => 4 = 5 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => 5 = 6 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => [2,6,5,4,3,1] => 4 = 5 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => 3 = 4 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [3,5,4,2,1] => 3 = 4 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,5,4,3] => 2 = 3 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 2 = 3 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => 3 = 4 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,2,1,5,4] => 2 = 3 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => 3 = 4 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => [5,4,3,2,6,1] => 4 = 5 - 1
[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] => [6,5,4,3,2,1,7] => 5 = 6 - 1
[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] => [1,8,7,6,5,4,3,2] => 6 = 7 - 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => ? ∊ {6,6} - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => [2,6,5,4,1,3] => 4 = 5 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5,6] => [3,6,5,4,2,1] => 4 = 5 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,5,1,4,3] => 3 = 4 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [3,5,4,1,2] => 3 = 4 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 3 = 4 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [3,1,5,4,2] => 2 = 3 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [3,2,5,1,4] => 2 = 3 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => 3 = 4 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,4,1,5,6] => [5,4,3,6,2,1] => 4 = 5 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => 3 = 4 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,1,5] => [5,4,3,1,6,2] => 4 = 5 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? ∊ {6,6} - 1
[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] => [7,6,5,4,3,2,1,8] => 6 = 7 - 1
[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] => [1,9,8,7,6,5,4,3,2] => 7 = 8 - 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,8] => [2,8,7,6,5,4,3,1] => ? ∊ {6,6,6,6,7} - 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => [2,7,6,5,4,1,3] => ? ∊ {6,6,6,6,7} - 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => ? ∊ {6,6,6,6,7} - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,4] => [2,6,5,1,4,3] => 4 = 5 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,1,2,3,6,5] => [3,6,5,4,1,2] => 4 = 5 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,5,6] => [4,6,5,3,2,1] => 4 = 5 - 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => 3 = 4 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [3,5,1,4,2] => 3 = 4 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => 3 = 4 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 3 = 4 - 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? ∊ {6,6,6,6,7} - 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? ∊ {6,6,6,6,7} - 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7,9] => [2,9,8,7,6,5,4,3,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => [2,8,7,6,5,4,1,3] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7,8] => [3,8,7,6,5,4,2,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [6,1,2,3,7,4,5] => [2,7,6,5,1,4,3] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => [3,7,6,5,4,1,2] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2,3,4,5,6,1,7,8] => [7,6,5,4,3,8,2,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,8,1,7] => [7,6,5,4,3,1,8,2] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1,9] => [8,7,6,5,4,3,2,9,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8} - 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8,10] => [2,10,9,8,7,6,5,4,3,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => [2,9,8,7,6,5,4,1,3] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,8,9] => [3,9,8,7,6,5,4,2,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [7,1,2,3,4,8,5,6] => [2,8,7,6,5,1,4,3] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,8,7] => [3,8,7,6,5,4,1,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6,7,8] => [4,8,7,6,5,3,2,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [6,1,2,7,3,4,5] => [2,7,6,1,5,4,3] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6] => [3,7,6,5,1,4,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,3,6,7,4] => [3,7,6,5,2,1,4] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,1,2,3,5,7,6] => [4,7,6,5,3,1,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,1,4,5,6,7] => [6,5,7,4,3,2,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,4,1,5,7,6] => [6,5,4,7,3,1,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6] => [6,5,4,1,7,3,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,1,8,7] => [7,6,5,4,3,8,1,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1,8,9] => [8,7,6,5,4,3,9,2,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,5,6,7,1,4] => [6,5,3,2,1,7,4] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,7,8,1,6] => [7,6,5,4,2,1,8,3] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,9,1,8] => [8,7,6,5,4,3,1,9,2] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1,10] => [9,8,7,6,5,4,3,2,10,1] => ? ∊ {6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9} - 1
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => [1,12,11,10,9,8,7,6,5,4,3,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9,11] => [2,11,10,9,8,7,6,5,4,3,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,10,8] => [2,10,9,8,7,6,5,4,1,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7,9,10] => [3,10,9,8,7,6,5,4,2,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,9,6,7] => [2,9,8,7,6,5,1,4,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,9,8] => [3,9,8,7,6,5,4,1,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7,8,9] => [4,9,8,7,6,5,3,2,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [7,1,2,3,8,4,5,6] => [2,8,7,6,1,5,4,3] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [6,1,2,3,4,8,5,7] => [3,8,7,6,5,1,4,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,7,8,5] => [3,8,7,6,5,2,1,4] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11} - 1

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000956
(load all 6 compositions to match this statistic)

Mapping

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: 42%●distinct values known / distinct values provided: 42%

Values

[1]
 => [1,0,1,0]
 => [2,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 5
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 4
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 4
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 5
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ? ∊ {6,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 5
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 4
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 4
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 4
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 5
[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] => ? ∊ {6,6}
[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] => ? ∊ {6,6,7,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? ∊ {6,6,7,7}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 5
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 5
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 4
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 4
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 4
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 4
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 5
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 5
[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] => ? ∊ {6,6,7,7}
[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] => ? ∊ {6,6,7,7}
[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] => ? ∊ {6,6,6,6,7,7,8,8}
[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] => ? ∊ {6,6,6,6,7,7,8,8}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? ∊ {6,6,6,6,7,7,8,8}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? ∊ {6,6,6,6,7,7,8,8}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 5
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 5
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 5
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 4
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 4
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => 5
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 3
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 4
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => 5
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => ? ∊ {6,6,6,6,7,7,8,8}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => ? ∊ {6,6,6,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ? ∊ {6,6,6,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => ? ∊ {6,6,6,6,7,7,8,8}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,1,5,6] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,7,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => ? ∊ {6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [7,3,4,1,2,5,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,1,5,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,1,6] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [6,2,3,4,5,7,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,7,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,7,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,7,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,7,1] => ? ∊ {5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}

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]].

Matching statistic: St000863

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000863: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 58%

Values

[1]
 => [1,0]
 => [1] => [1] => 1
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 2
[1,1]
 => [1,1,0,0]
 => [2,1] => [2,1] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => 3
[2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => 3
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => 3
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => 4
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,5,2,3,4] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [5,1,2,3,6,4] => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [1,3,4,2] => 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [4,5,1,2,3] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [4,1,2,5,6,3] => 4
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [3,4,1,5,2] => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [3,1,4,5,6,2] => 5
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? ∊ {4,5,5,6,6,7}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 5
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [6,1,2,3,4,7,5] => ? ∊ {4,5,5,6,6,7}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,5,3] => 3
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [5,6,1,2,3,4] => 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [5,1,2,3,6,7,4] => ? ∊ {4,5,5,6,6,7}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,4,5,1,2] => 4
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,5,3,4] => 4
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [4,5,1,2,6,3] => 4
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [4,1,2,5,6,7,3] => ? ∊ {4,5,5,6,6,7}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [4,1,5,2,3] => 3
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [3,4,1,5,6,2] => 5
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? ∊ {4,5,5,6,6,7}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? ∊ {4,5,5,6,6,7}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [8,1,2,3,4,5,6,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [7,1,2,3,4,5,8,6] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [1,5,2,3,6,4] => 4
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,4,7,5] => [6,7,1,2,3,4,5] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [6,1,2,3,4,7,8,5] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [1,3,4,5,2] => 4
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => [4,5,1,6,2,3] => 4
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 5
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,3,6,7,4] => [5,6,1,2,3,7,4] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [5,1,2,3,6,7,8,4] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [1,3,2,5,4] => 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => [3,4,5,1,6,2] => 5
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => [5,1,6,2,3,4] => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,6,7,3] => [4,5,1,2,6,7,3] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [4,1,2,5,6,7,8,3] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [1,2,4,5,3] => 4
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => [4,1,5,2,6,3] => 4
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,2] => [3,4,1,5,6,7,2] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,2] => [3,1,4,5,6,7,8,2] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,8,1] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,9,8] => [9,1,2,3,4,5,6,7,8] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,6,7] => [1,8,2,3,4,5,6,7] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,7] => [8,1,2,3,4,5,6,9,7] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,4,5] => [1,6,2,3,4,7,5] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,5,8,6] => [7,8,1,2,3,4,5,6] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,6] => [7,1,2,3,4,5,8,9,6] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,4,5,6,2,3] => [1,4,2,5,6,3] => 4
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,3,7,4] => [5,6,1,2,7,3,4] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => 6
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,4,7,8,5] => [6,7,1,2,3,4,8,5] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,5] => [6,1,2,3,4,7,8,9,5] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [3,4,5,6,1,2] => 5
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,5,2,6,7,3] => [4,5,1,6,2,7,3] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,3,4,7,5] => [6,1,7,2,3,4,5] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,5,3,6,7,8,4] => [5,6,1,2,3,7,8,4] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,9,4] => [5,1,2,3,6,7,8,9,4] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [3,4,1,5,6,7,2] => [3,4,5,1,6,7,2] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,2,3,6,7,4] => [5,1,6,2,3,7,4] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,2,5,6,7,8,3] => [4,5,1,2,6,7,8,3] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,9,3] => [4,1,2,5,6,7,8,9,3] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [4,1,2,5,6,7,3] => [4,1,5,2,6,7,3] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,8,2] => [3,4,1,5,6,7,8,2] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,2] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,9,1] => ? ∊ {3,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8,8,9,9}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9] => [10,1,2,3,4,5,6,7,8,9] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,9,7,8] => [1,9,2,3,4,5,6,7,8] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,7,9,10,8] => [9,1,2,3,4,5,6,7,10,8] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,5,6] => [1,7,2,3,4,5,8,6] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,5,8,6,9,7] => [8,9,1,2,3,4,5,6,7] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,10,7] => [8,1,2,3,4,5,6,9,10,7] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,4,8,5] => [6,7,1,2,3,8,4,5] => ? ∊ {4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}

Description

The length of the first row of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the length of the first row of

$P$ and

$Q$ .

Matching statistic: St000786

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 58%

Values

[1]
 => [1,0]
 => [[]]
 => ([(0,1)],2)
 => 1
[2]
 => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[1,1]
 => [1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2
[3]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[2,2]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 4
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 5
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,6,6}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 5
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[],[[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,6,6}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[],[],[[[]],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,6,6}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [[],[[[]],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 4
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[],[[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,6,6}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[[[]],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 5
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,6,6}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 7
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[],[]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[],[],[[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[],[[[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[],[],[[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[],[],[[[],[]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 5
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [[],[],[],[[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[],[],[[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[],[[[],[]],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 4
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[],[],[[[[]]]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 4
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [[],[],[[[]],[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[],[],[[],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[[[],[]],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 4
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[],[[[[]]],[]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [[],[[[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[],[[],[],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[[[[]]],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 5
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[[[]],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[],[],[]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[],[],[],[[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[],[],[[[]]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,7),(7,8)],9)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[],[],[],[[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [[],[],[],[[[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [[],[],[],[],[[[]],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,7),(7,8)],9)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[],[],[],[[],[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(6,8),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[],[[[],[],[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 5
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [[],[],[[[],[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[],[],[],[[[[]]]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [[],[],[],[[[]],[],[]]]
 => ([(0,8),(1,8),(2,7),(3,7),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[],[],[],[[],[],[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,8),(5,8),(6,8),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[[[],[],[]],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 4
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [[],[[[],[]],[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [[],[],[[[[]]],[]]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [[],[],[[[]],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[],[],[[],[],[],[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(6,8),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [[[[],[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [[],[[[[]]],[],[]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[],[[[]],[],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[],[[],[],[],[],[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [[[[[]]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[]],[],[],[],[],[]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[],[],[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[],[],[],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[],[],[],[]]
 => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[],[],[],[],[[]]]
 => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(9,10)],11)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[],[],[],[[[]]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(8,9)],10)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[],[],[],[],[[],[]]]
 => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [[],[],[],[],[[[],[]]]]
 => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(6,8)],9)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [[],[],[],[],[],[[[]],[]]]
 => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(6,7),(7,8),(8,9)],10)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[],[],[],[],[[],[],[]]]
 => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,9),(7,9),(8,9),(9,10)],11)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [[],[],[[[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [[],[],[],[[[],[]],[]]]
 => ([(0,8),(1,8),(2,8),(3,6),(4,6),(5,7),(6,7),(7,8)],9)
 => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}

Description

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

Matching statistic: St001566

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 58%

Values

[1]
 => [1,0]
 => [1,0]
 => [1] => 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 2
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 4
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 5
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => ? ∊ {4,5,5,6,6,7}
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ? ∊ {4,5,5,6,6,7}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 5
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? ∊ {4,5,5,6,6,7}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 4
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7] => ? ∊ {4,5,5,6,6,7}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 3
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [3,2,4,1,5,6] => 4
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7] => ? ∊ {4,5,5,6,6,7}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 4
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6] => 5
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7] => ? ∊ {4,5,5,6,6,7}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => 7
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [8,7,6,5,4,3,2,1] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [6,5,4,3,2,1,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => 5
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [5,4,3,2,6,1,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [5,4,3,2,1,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 4
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => 4
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => 4
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [4,3,2,5,1,6,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [3,4,2,1,5,6] => 4
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,2,4,5,1,6] => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [3,2,4,1,5,6,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,1,5,6] => 5
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [9,8,7,6,5,4,3,2,1] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,7,6,5,4,3,2,1,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,6,5,4,3,2,8,1] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [7,6,5,4,3,2,1,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [6,5,4,3,2,7,1,8] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,4,6,3,2,1] => 5
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [5,4,3,6,2,1,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,4,3,2,6,7,1] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [5,4,3,2,6,1,7,8] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1,6,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => 4
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,3,5,2,6,1] => 4
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [4,3,5,2,1,6,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [4,3,2,5,6,1,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [4,3,2,5,1,6,7,8] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [3,4,2,1,5,6,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [3,2,4,5,1,6,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [3,2,4,1,5,6,7,8] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,5,6,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10,9,8,7,6,5,4,3,2,1] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,8,7,6,5,4,3,2,1,10] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [8,7,6,5,4,3,2,9,1] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1,9,10] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [7,6,5,4,3,8,2,1] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
 => [7,6,5,4,3,2,8,1,9] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1,8,9,10] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}

Description

The length of the longest arithmetic progression in a permutation. For a permutation

$\pi$ of length

$n$ , this is the biggest

$k$ such that there exist

$1 \leq i_1 < \dots < i_k \leq n$ with

$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$

Matching statistic: St000923
(load all 2 compositions to match this statistic)

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%

Values

[1]
 => [1,0]
 => [1,0]
 => [1] => ? = 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 2
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [2,1] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 4
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [6,5,4,1,2,3] => 4
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => 5
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ? ∊ {4,5,5,6,6,7}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,4,6] => 5
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => ? ∊ {4,5,5,6,6,7}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 4
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5,1,2,3,4] => ? ∊ {4,5,5,6,6,7}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 3
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1,2,4] => 4
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,1,2,3] => ? ∊ {4,5,5,6,6,7}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 4
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => 5
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => ? ∊ {4,5,5,6,6,7}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => ? ∊ {4,5,5,6,6,7}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,5,7] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [8,7,1,2,3,4,5,6] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 5
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [8,7,6,1,2,3,4,5] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 4
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 4
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,2,3,6] => 4
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [7,6,4,1,2,3,5] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,1,2,3,4] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1,3,4] => 4
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1,2,5] => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1,2,4] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,1,2,3] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 4
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,4] => 5
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,3] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,1,2] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => ? ∊ {4,5,5,5,5,6,6,6,6,7,7,8,8}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [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] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,1,2,3,4,5,6,8] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [9,8,1,2,3,4,5,6,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [5,1,2,3,4,6,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => [9,8,7,1,2,3,4,5,6] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,2,4,5,6] => 5
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [8,7,5,1,2,3,4,6] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,1,2,3,4,5] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 4
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,2,4,6] => 4
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [7,5,4,1,2,3,6] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,1,2,3,5] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,1,2,3,4] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 3
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,1,3,4] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,1,2,5] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,3,1,2,4] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,1,2,3] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1,4] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,2,1,3] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,1,2] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1] => ? ∊ {4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [8,1,2,3,4,5,6,7,9] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [10,9,1,2,3,4,5,6,7,8] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [6,1,2,3,4,5,7,8] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
 => [9,7,1,2,3,4,5,6,8] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => [10,9,8,1,2,3,4,5,6,7] => ? ∊ {4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10}

Description

The minimal number with no two order isomorphic substrings of this length in a permutation. For example, the length

$3$ substrings of the permutation

$12435$ are

$124$ ,

$243$ and

$435$ , whereas its length

$2$ substrings are

$12$ ,

$24$ ,

$43$ and

$35$ . No two sequences among

$124$ ,

$243$ and

$435$ are order isomorphic, but

$12$ and

$24$ are, so the statistic on

$12435$ is

$3$ . This is inspired by [[St000922]].

Matching statistic: St001232
(load all 14 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 83%

Values

[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {2,3,4}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,3,4}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,4}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,5}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {3,3,5}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,5}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,4,4,4,5,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,4,4,4,4,5,5,7,7}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => 6
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,6,7,7,8,8}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => 7
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 8
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,8,9,9}
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 7
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 7
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 10
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => 8
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => 8
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 6
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 9
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 8
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => 9
[6,4,2]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 7

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001655

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 58%

Values

[1]
 => [[1]]
 => [1] => ([],1)
 => 1
[2]
 => [[1,2]]
 => [1,2] => ([],2)
 => 2
[1,1]
 => [[1],[2]]
 => [2,1] => ([(0,1)],2)
 => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => 3
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([],4)
 => 4
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([],5)
 => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([],6)
 => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 3
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([],7)
 => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 6
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 5
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => 4
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 4
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 4
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,8,8}
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ([],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[3,1,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}
[2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,9,9}

Description

The general position number of a graph. A set

$S$ of vertices in a graph

$G$ is a general position set if no three vertices of

$S$ lie on a shortest path between any two of them.

The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001656The monophonic position number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000213The number of weak exceedances (also weak excedences) of a permutation.