Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000662
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1] => 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => [1,2,3] => 0
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [2,4,1,3] => [1,3,2,4] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,4,2,1] => [1,2,3,4] => 0
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => [1,2,4,3] => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => [1,4,2,3] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,4,5,1,3] => [1,3,2,4,5] => 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,5,1,2] => [1,2,3,4,5] => 0
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3,5,1,4] => [1,4,2,3,5] => 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,4,5,2,1] => [1,2,3,4,5] => 0
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,5,3,1] => [1,2,4,5,3] => 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,2,3] => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,3,5,4,1] => [1,2,3,5,4] => 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,4,2] => [1,3,5,2,4] => 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,5,2,4,1] => [1,2,4,3,5] => 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,5,2,3,1] => [1,2,3,4,5] => 0
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,1,3,2] => [1,3,2,4,5] => 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,5,1,4,3] => [1,4,2,5,3] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [3,5,1,4,2] => [1,4,2,3,5] => 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,5,3,2,1] => [1,2,3,4,5] => 0
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,5,4,2,1] => [1,2,3,5,4] => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,3,1] => [1,2,5,3,4] => 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [2,4,5,6,1,3] => [1,3,2,4,5,6] => 1
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1] => [1] => ([],1)
 => ? = 0 - 1
[[1,2]]
 => [2] => [1,1] => ([(0,1)],2)
 => -1 = 0 - 1
[[1],[2]]
 => [2] => [1,1] => ([(0,1)],2)
 => -1 = 0 - 1
[[1,2,3]]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 0 - 1
[[1,3],[2]]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 0 - 1
[[1,2],[3]]
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[1],[2],[3]]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 0 - 1
[[1,2,3,4]]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[[1,3,4],[2]]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[[1,2,4],[3]]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
[[1,2,3],[4]]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[[1,4],[2],[3]]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[[1,3],[2],[4]]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[[1,2,3,4,5]]
 => [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 = 0 - 1
[[1,3,4,5],[2]]
 => [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 = 0 - 1
[[1,2,4,5],[3]]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2,3,5],[4]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2,3,4],[5]]
 => [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)
 => 0 = 1 - 1
[[1,3,5],[2,4]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2,5],[3,4]]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[[1,3,4],[2,5]]
 => [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)
 => 0 = 1 - 1
[[1,2,4],[3,5]]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2,3],[4,5]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[1,4,5],[2],[3]]
 => [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 = 0 - 1
[[1,3,5],[2],[4]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2,5],[3],[4]]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[1,3,4],[2],[5]]
 => [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)
 => 0 = 1 - 1
[[1,2,4],[3],[5]]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2,3],[4],[5]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,4],[2,5],[3]]
 => [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)
 => 0 = 1 - 1
[[1,3],[2,5],[4]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[[1,2],[3,5],[4]]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,3],[2,4],[5]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[[1,2],[3,4],[5]]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,5],[2],[3],[4]]
 => [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 = 0 - 1
[[1,4],[2],[3],[5]]
 => [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)
 => 0 = 1 - 1
[[1,3],[2],[4],[5]]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1,2],[3],[4],[5]]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[[1],[2],[3],[4],[5]]
 => [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 = 0 - 1
[[1,2,3,4,5,6]]
 => [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 = 0 - 1
[[1,3,4,5,6],[2]]
 => [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 = 0 - 1
[[1,2,4,5,6],[3]]
 => [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 - 1
[[1,2,3,5,6],[4]]
 => [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 - 1
[[1,2,3,4,6],[5]]
 => [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 - 1
[[1,2,3,4,5],[6]]
 => [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)
 => 0 = 1 - 1
[[1,3,5,6],[2,4]]
 => [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 - 1
[[1,2,5,6],[3,4]]
 => [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)
 => ? = 0 - 1
[[1,3,4,6],[2,5]]
 => [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 - 1
[[1,2,4,6],[3,5]]
 => [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)
 => ? = 2 - 1
[[1,2,3,6],[4,5]]
 => [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)
 => ? = 2 - 1
[[1,3,4,5],[2,6]]
 => [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)
 => 0 = 1 - 1
[[1,2,4,5],[3,6]]
 => [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 - 1
[[1,2,3,5],[4,6]]
 => [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 - 1
[[1,2,3,4],[5,6]]
 => [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)
 => ? = 2 - 1
[[1,4,5,6],[2],[3]]
 => [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 = 0 - 1
[[1,3,5,6],[2],[4]]
 => [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 - 1
[[1,2,5,6],[3],[4]]
 => [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)
 => ? = 2 - 1
[[1,3,4,6],[2],[5]]
 => [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 - 1
[[1,2,4,6],[3],[5]]
 => [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)
 => ? = 2 - 1
[[1,2,3,6],[4],[5]]
 => [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)
 => ? = 2 - 1
[[1,3,4,5],[2],[6]]
 => [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)
 => 0 = 1 - 1
[[1,2,4,5],[3],[6]]
 => [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 - 1
[[1,2,3,5],[4],[6]]
 => [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 - 1
[[1,2,3,4],[5],[6]]
 => [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 - 1
[[1,3,5],[2,4,6]]
 => [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 - 1
[[1,2,5],[3,4,6]]
 => [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 - 1
[[1,3,4],[2,5,6]]
 => [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 - 1
[[1,2,4],[3,5,6]]
 => [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 - 1
[[1,2,3],[4,5,6]]
 => [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)
 => ? = 0 - 1
[[1,4,6],[2,5],[3]]
 => [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 - 1
[[1,3,6],[2,5],[4]]
 => [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)
 => ? = 0 - 1
[[1,2,6],[3,5],[4]]
 => [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 - 1
[[1,3,6],[2,4],[5]]
 => [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)
 => ? = 2 - 1
[[1,2,6],[3,4],[5]]
 => [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 - 1
[[1,4,5],[2,6],[3]]
 => [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)
 => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
 => [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 - 1
[[1,5,6],[2],[3],[4]]
 => [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 = 0 - 1
[[1,4,5],[2],[3],[6]]
 => [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)
 => 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
 => [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)
 => 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
 => [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 = 0 - 1
[[1,5],[2],[3],[4],[6]]
 => [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)
 => 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
 => [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 = 0 - 1
[[1,2,3,4,5,6,7]]
 => [7] => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 0 - 1
[[1,3,4,5,6,7],[2]]
 => [7] => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 0 - 1
[[1,2,3,4,5,6],[7]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,4,5,6,7],[2],[3]]
 => [7] => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 0 - 1
[[1,3,4,5,6],[2],[7]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,4,5,6],[2,7],[3]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,5,6,7],[2],[3],[4]]
 => [7] => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 0 - 1
[[1,4,5,6],[2],[3],[7]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,5,6],[2,7],[3],[4]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,6,7],[2],[3],[4],[5]]
 => [7] => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 0 - 1
[[1,5,6],[2],[3],[4],[7]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,6],[2,7],[3],[4],[5]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 1 - 1
[[1,7],[2],[3],[4],[5],[6]]
 => [7] => [1,1,1,1,1,1,1] => ([(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)
 => -1 = 0 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [6,1] => [2,1,1,1,1,1] => ([(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)
 => 0 = 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.