Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001061
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Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,2]]
 => [1,2] => 0
[2,1] => [[1],[2]]
 => [2,1] => 1
[1,2,3] => [[1,2,3]]
 => [1,2,3] => 0
[1,3,2] => [[1,2],[3]]
 => [3,1,2] => 0
[2,1,3] => [[1,3],[2]]
 => [2,1,3] => 1
[2,3,1] => [[1,2],[3]]
 => [3,1,2] => 0
[3,1,2] => [[1,3],[2]]
 => [2,1,3] => 1
[3,2,1] => [[1],[2],[3]]
 => [3,2,1] => 2
[1,2,3,4] => [[1,2,3,4]]
 => [1,2,3,4] => 0
[1,2,4,3] => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,3,2,4] => [[1,2,4],[3]]
 => [3,1,2,4] => 0
[1,3,4,2] => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,4,2,3] => [[1,2,4],[3]]
 => [3,1,2,4] => 0
[1,4,3,2] => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[2,1,3,4] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[2,1,4,3] => [[1,3],[2,4]]
 => [2,4,1,3] => 0
[2,3,1,4] => [[1,2,4],[3]]
 => [3,1,2,4] => 0
[2,3,4,1] => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[2,4,1,3] => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[2,4,3,1] => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[3,1,2,4] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[3,1,4,2] => [[1,3],[2,4]]
 => [2,4,1,3] => 0
[3,2,1,4] => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[3,2,4,1] => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
[3,4,1,2] => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[3,4,2,1] => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[4,1,2,3] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[4,1,3,2] => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
[4,2,1,3] => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[4,2,3,1] => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
[4,3,1,2] => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[4,3,2,1] => [[1],[2],[3],[4]]
 => [4,3,2,1] => 3
[1,2,3,4,5] => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 0
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 0
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 0
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 0
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,3,5,4,2] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 0
[1,4,2,5,3] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,4,5,3,2] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0
Description
The number of indices that are both descents and recoils of a permutation.
Matching statistic: St000454
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => ([],2)
 => 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0
[3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,2,4,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,3,2,4] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,3,4,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,1,3,4] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,3,1,4] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[2,3,4,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,4,1,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,4,3,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,1,2,4] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,1,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3,4,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,4,2,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,2,3] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,1,3,2] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,1,3] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,3,1,2] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,2,4,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[2,1,3,4,5] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,1,4,3,5] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,4,3,2,1] => [5,4,3,2,1] => [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)
 => 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [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)
 => 4
[6,4,3,2,1,5] => [6,4,3,2,1,5] => [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)
 => 4
[6,5,3,2,1,4] => [6,5,3,2,1,4] => [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)
 => 4
[6,5,4,2,1,3] => [6,5,4,2,1,3] => [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)
 => 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [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)
 => 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [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)
 => 5
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.