Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001593
Mapping
St001593: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 0
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 2
[2,2]
 => 0
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 1
[4,1]
 => 3
[3,2]
 => 2
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 1
[5,1]
 => 4
[4,2]
 => 5
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 2
[3,1,1,1]
 => 0
[2,2,2]
 => 0
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 1
[6,1]
 => 5
[5,2]
 => 9
[5,1,1]
 => 0
[4,3]
 => 5
[4,2,1]
 => 7
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 1
[7,1]
 => 6
[6,2]
 => 14
[6,1,1]
 => 0
[5,3]
 => 14
[5,2,1]
 => 16
Description
This is the number of standard Young tableaux of the given shifted shape. For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e., $$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$ In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St001498
(load all 8 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 22%●distinct values known / distinct values provided: 1%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => ? = 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => ? = 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1}
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {1,1}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,2}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,2}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,2,3}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,2,3}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,2,3}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,2,3}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,1,2,4,5}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,2,4,5}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,2,4,5}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,2,4,5}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,2,4,5}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 0
[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] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[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] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 0
[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] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 0
[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] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,1,5,5,7,9}
[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] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[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] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[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] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 0
[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] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 0
[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] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[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] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,1,6,12,14,14,16}
[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] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[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] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[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] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[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] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[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] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[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] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 0
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,1,2] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 0
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 0
[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] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[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] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => 0
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 0
[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] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 0
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 0
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001629
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 2% ●values known / values provided: 14%●distinct values known / distinct values provided: 2%
Values
[1]
 => [1,0]
 => [1] => [1] => ? = 1
[2]
 => [1,0,1,0]
 => [1,1] => [2] => ? ∊ {0,1}
[1,1]
 => [1,1,0,0]
 => [2] => [1] => ? ∊ {0,1}
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => [3] => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,2] => [1,1] => ? ∊ {0,1}
[1,1,1]
 => [1,1,0,1,0,0]
 => [3] => [1] => ? ∊ {0,1}
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => 0
[2,2]
 => [1,1,1,0,0,0]
 => [3] => [1] => ? ∊ {0,0,2}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => [1,1] => ? ∊ {0,0,2}
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [4] => [1] => ? ∊ {0,0,2}
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => 0
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => ? ∊ {0,0,2,3}
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => 0
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [4] => [1] => ? ∊ {0,0,2,3}
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,2,3}
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => [1] => ? ∊ {0,0,2,3}
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [6] => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [4,1] => 0
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1] => 0
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4] => [1] => ? ∊ {0,0,0,2,4,5}
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,0,2,4,5}
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1] => ? ∊ {0,0,0,2,4,5}
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => [1] => ? ∊ {0,0,0,2,4,5}
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,2,4,5}
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,2,4,5}
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => [7] => 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => [5,1] => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1] => 0
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => [4,1] => 0
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,0,0,5,5,7,9}
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => [3,1] => 0
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => [1] => ? ∊ {0,0,0,0,5,5,7,9}
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,0,0,5,5,7,9}
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,0,5,5,7,9}
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => [1] => ? ∊ {0,0,0,0,5,5,7,9}
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,0,5,5,7,9}
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,6] => [1,1] => ? ∊ {0,0,0,0,5,5,7,9}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7] => [1] => ? ∊ {0,0,0,0,5,5,7,9}
[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] => [8] => 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => [6,1] => 0
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => [4,1] => 0
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,3] => [5,1] => 0
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => [2,1] => 0
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => [3,1] => 0
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => [2,1] => 0
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,6] => [2,1] => 0
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [7] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,7] => [1,1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8] => [1] => ? ∊ {0,0,0,0,0,0,6,12,14,14,16}
[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] => [9] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,2] => [7,1] => 0
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => [5,1] => 0
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => [6,1] => 0
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => [3,1] => 0
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,4] => [5,1] => 0
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => [3,1] => 0
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,5] => [4,1] => 0
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,6] => [2,1] => 0
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,6] => [3,1] => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [7] => [1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,7] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[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]
 => [1,1,7] => [2,1] => 0
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [6] => [1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [7] => [1] => ? ∊ {0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,3] => [6,1] => 0
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,3] => [7,1] => 0
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,4] => [5,1] => 0
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,4] => [6,1] => 0
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => [2,1] => 0
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 2%●distinct values known / distinct values provided: 1%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => ? = 1
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? ∊ {0,1}
[1,1]
 => [[1],[2]]
 => [2] => ([],2)
 => ? ∊ {0,1}
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? ∊ {1,1}
[2,1]
 => [[1,2],[3]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 0
[1,1,1]
 => [[1],[2],[3]]
 => [3] => ([],3)
 => ? ∊ {1,1}
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? ∊ {1,2}
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4] => ([],4)
 => ? ∊ {1,2}
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? ∊ {1,2,3}
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,3}
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => ? ∊ {1,2,3}
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? ∊ {0,1,2,4,5}
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,1,2,4,5}
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,1,2,4,5}
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,1,2,4,5}
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 0
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => ? ∊ {0,1,2,4,5}
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => 0
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,5] => ([(4,6),(5,6)],7)
 => 0
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7] => ([],7)
 => ? ∊ {0,0,0,1,5,5,7,9}
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [8] => ([],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [5,2,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [3,2,2,1] => ([(0,7),(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8] => ([],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,6,12,14,14,16}
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [9] => ([],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,7,12,14,20,28,30,42}
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.