Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000902
(load all 91 compositions to match this statistic)
Mapping
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[2,3] => 1
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 2
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000900
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 1
[1,1] => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[2] => [2]
 => [[1,2]]
 => [2] => 1
[1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[1,2] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[2,1] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[3] => [3]
 => [[1,2,3]]
 => [3] => 1
[1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 4
[1,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,2,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,3] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[2,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,1] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[4] => [4]
 => [[1,2,3,4]]
 => [4] => 1
[1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 5
[1,1,1,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,2,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,2,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,2,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[1,3,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[2,1,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 1
[3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 1
[4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[5] => [5]
 => [[1,2,3,4,5]]
 => [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 6
[1,1,1,1,2] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,3] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,2,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,1,3,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[1,1,4] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,2,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,2,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,2,3] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[1,3,2] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[1,4,1] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 1
[1,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[2,1,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[2,1,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
Description
The minimal number of repetitions of a part in an integer composition. This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000617
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? = 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 3
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
 => ? = 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
 => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => ? = 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => 7
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
 => ? = 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => 8
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => 9
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
Mapping
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 11%
Values
[1] => [1] => ([],1)
 => ([],2)
 => ? = 1 - 1
[1,1] => [2] => ([],2)
 => ([],3)
 => ? = 2 - 1
[2] => [1,1] => ([(0,1)],2)
 => ([(1,2)],3)
 => 0 = 1 - 1
[1,1,1] => [3] => ([],3)
 => ([],4)
 => ? = 3 - 1
[1,2] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 0 = 1 - 1
[2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,1,1] => [4] => ([],4)
 => ([],5)
 => ? = 4 - 1
[1,1,2] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 0 = 1 - 1
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,1,1,1] => [5] => ([],5)
 => ([],6)
 => ? = 5 - 1
[1,1,1,2] => [1,4] => ([(3,4)],5)
 => ([(4,5)],6)
 => 0 = 1 - 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => ([],6)
 => ([],7)
 => ? = 6 - 1
[1,1,1,1,2] => [1,5] => ([(4,5)],6)
 => ([(5,6)],7)
 => 0 = 1 - 1
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,3] => [1,1,2,1,1] => ([(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)
 => ([(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)
 => ? = 2 - 1
[4,1,1] => [3,1,1,1] => ([(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)
 => ([(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)
 => 0 = 1 - 1
[4,2] => [1,2,1,1,1] => ([(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)
 => ([(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)
 => ? = 1 - 1
[5,1] => [2,1,1,1,1] => ([(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)
 => ([(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)
 => 0 = 1 - 1
[6] => [1,1,1,1,1,1] => ([(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)
 => ([(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)
 => 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => ([],7)
 => ([],8)
 => ? = 7 - 1
[1,1,1,1,1,2] => [1,6] => ([(5,6)],7)
 => ([(6,7)],8)
 => ? = 1 - 1
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
 => ([(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,1,1,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,1,5] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.