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Matching statistic: St000887
St000887: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 3
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The maximal number of nonzero entries on a diagonal of a permutation matrix.
For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right.
In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$
Matching statistic: St000888
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000888: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000888: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,0],[0,1]]
=> 2
[2,1] => [[0,1],[1,0]]
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 1
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 1
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 2
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 2
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 2
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 2
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 2
Description
The maximal sum of entries on a diagonal of an alternating sign matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr}
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & -1 & 1 \\
0 & 0 & 1 & 0
\end{array}\right)$$
are $(0,1,1,0,1,1,0)$, so the statistic is $1$.
This is a natural extension of [[St000887]] to alternating sign matrices.
Matching statistic: St000892
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000892: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000892: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,0],[0,1]]
=> 2
[2,1] => [[0,1],[1,0]]
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 1
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 1
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 2
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 2
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 2
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 2
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 2
Description
The maximal number of nonzero entries on a diagonal of an alternating sign matrix.
For example, for the matrix $$\left(\begin{array}{rrrr}
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & -1 & 1 \\
0 & 0 & 1 & 0
\end{array}\right)$$
the numbers of nonzero entries are $(0,1,1,2,1,1,0)$, so the statistic is $2$.
This is a natural extension of [[St000887]] to alternating sign matrices. See [[St000888]] for the maximal sums.
Matching statistic: St001165
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 74%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 74%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1,0]
=> 1
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 2
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,3}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,2,2,2,4}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,2,2,2,4}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,2,2,2,4}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,2,2,2,4}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,2,2,2,4}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,2,2,2,4}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 3
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5}
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000741
Values
[1,2] => ([],2)
=> ([],3)
=> 1
[2,1] => ([(0,1)],2)
=> ([(1,2)],3)
=> ? = 2
[1,2,3] => ([],3)
=> ([],4)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(2,3)],4)
=> ? ∊ {2,3}
[2,1,3] => ([(1,2)],3)
=> ([(2,3)],4)
=> ? ∊ {2,3}
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4] => ([],4)
=> ([],5)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[1,3,2,4] => ([(2,3)],4)
=> ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,4}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,3,2,1] => ([(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)
=> 3
[1,2,3,4,5] => ([],5)
=> ([],6)
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,2,4,3,5] => ([(3,4)],5)
=> ([(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,4,3,2] => ([(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)
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> ([(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[2,4,5,3,1] => ([(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)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,4,5,2,1] => ([(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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[5,2,3,4,1] => ([(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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
[5,4,1,2,3] => ([(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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,5}
Description
The Colin de Verdière graph invariant.
Matching statistic: St000259
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 2
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,3}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,3}
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,3}
[2,3,1] => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[2,3,1,4] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,4,1] => [2,4,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => [2,4,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,4}
[3,2,4,1] => [3,2,4,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3,4,2,1] => [3,4,2,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4,1,3,2] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [4,1,3,2] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [4,2,1,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[4,3,1,2] => [4,3,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,5,2,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,1,3,4,5] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,1,4,3,5] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[2,3,1,4,5] => [2,5,1,4,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,5,4] => [2,5,1,4,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,4,1,5] => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,4,5,1] => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,5,1,4] => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,5,4,1] => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [2,5,1,4,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,1,5,3] => [2,5,1,4,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,3,1,5] => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,3,5,1] => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,5,1,3] => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,5,3,1] => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,1,3,4] => [2,5,1,4,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,1,4,3] => [2,5,1,4,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,3,1,4] => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,3,4,1] => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,4,1,3] => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,4,5] => [3,1,5,4,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => [3,1,5,4,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,2,5] => [3,1,5,4,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,5,2] => [3,1,5,4,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,2,4] => [3,1,5,4,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [3,1,5,4,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
[3,2,4,1,5] => [3,2,5,1,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,4,5,1] => [3,2,5,4,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [3,2,5,1,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,5,4,1] => [3,2,5,4,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [3,5,1,4,2] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,5,2] => [3,5,1,4,2] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,2,1,5] => [3,5,2,1,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,5}
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001568
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 67%●distinct values known / distinct values provided: 40%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 67%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,1]
=> [2]
=> []
=> ? ∊ {1,2}
[2,1] => [2]
=> [1,1]
=> [1]
=> ? ∊ {1,2}
[1,2,3] => [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,3}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,3}
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1,3}
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,3,3,4}
[1,2,4,3] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {1,3,3,4}
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,2,1,4] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? ∊ {1,3,3,4}
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? ∊ {1,3,3,4}
[1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> [2]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,2,5,3,4] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,4,2,3,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,4,5,2,3] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,5,3,2,4] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,1,5,3,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,1,5,4,3] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,1,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,3,4,5,1] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,3,5,1,4] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,3,5,4,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,4,1,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,4,1,5,3] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,4,3,1,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,3,5,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,4,5,1,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,5,3,1] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,5,3,4,1] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,4,3,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,1,2,4,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,1,2,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,2,4,1,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,2,5,4,1] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,1,5,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,5,2,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,1,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,4,1,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,1,3,2,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,1,5,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,2,1,3,5] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,2,3,5,1] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,3,2,5,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,3,5,1,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,1,3,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,2,1,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,3,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,4,3,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,2,1,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,2,3,1,4] => [3,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,3,2,1,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,3,4,2,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,4,1,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,4,2,3,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000260
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> ([],2)
=> ? = 2
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,2,3}
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,2,3}
[3,2,1] => ([],3)
=> ([],3)
=> ? ∊ {2,2,3}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,4}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000771
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Values
[1,2] => ([],2)
=> ([],2)
=> ? = 2
[2,1] => ([(0,1)],2)
=> ([],1)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> ? ∊ {1,2,3}
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,2,3}
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,2,3}
[2,3,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,4}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000668
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 58%●distinct values known / distinct values provided: 40%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 58%●distinct values known / distinct values provided: 40%
Values
[1,2] => [2] => [2]
=> []
=> ? ∊ {1,2}
[2,1] => [2] => [2]
=> []
=> ? ∊ {1,2}
[1,2,3] => [3] => [3]
=> []
=> ? ∊ {1,1,1,2,2,3}
[1,3,2] => [1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,3}
[2,1,3] => [3] => [3]
=> []
=> ? ∊ {1,1,1,2,2,3}
[2,3,1] => [3] => [3]
=> []
=> ? ∊ {1,1,1,2,2,3}
[3,1,2] => [3] => [3]
=> []
=> ? ∊ {1,1,1,2,2,3}
[3,2,1] => [2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,3}
[1,2,3,4] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[1,2,4,3] => [2,2] => [2,2]
=> [2]
=> 2
[1,3,2,4] => [1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[1,3,4,2] => [1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[1,4,2,3] => [1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[1,4,3,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[2,1,4,3] => [2,2] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[2,3,4,1] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[2,4,1,3] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[2,4,3,1] => [3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[3,1,2,4] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[3,1,4,2] => [2,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,2] => [2,2]
=> [2]
=> 2
[3,2,4,1] => [2,2] => [2,2]
=> [2]
=> 2
[3,4,1,2] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[3,4,2,1] => [3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[4,1,2,3] => [4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[4,1,3,2] => [3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[4,2,1,3] => [2,2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[4,3,1,2] => [1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4}
[4,3,2,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,2,3,5,4] => [3,2] => [3,2]
=> [2]
=> 2
[1,2,4,3,5] => [2,3] => [3,2]
=> [2]
=> 2
[1,2,4,5,3] => [2,3] => [3,2]
=> [2]
=> 2
[1,2,5,3,4] => [2,3] => [3,2]
=> [2]
=> 2
[1,2,5,4,3] => [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[1,3,2,4,5] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,3,2,5,4] => [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,3,4,5,2] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,3,5,2,4] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,4,2,5,3] => [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,3,2,5] => [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,3,5,2] => [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,4,5,2,3] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,1,3,5,4] => [3,2] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [2,3] => [3,2]
=> [2]
=> 2
[2,1,4,5,3] => [2,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [2,3] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,1,5,4] => [3,2] => [3,2]
=> [2]
=> 2
[2,3,4,1,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,4,5,1] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,5,1,4] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,3,5,4,1] => [4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,1,3,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,1,5,3] => [3,2] => [3,2]
=> [2]
=> 2
[2,4,3,1,5] => [3,2] => [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [3,2] => [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,4,5,3,1] => [4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,1,3,4] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,1,4,3] => [4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,3,1,4] => [3,2] => [3,2]
=> [2]
=> 2
[2,5,3,4,1] => [4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[2,5,4,1,3] => [2,3] => [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[3,1,2,4,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,1,2,5,4] => [3,2] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [2,3] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [2,3] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [2,3] => [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[3,2,1,4,5] => [2,3] => [3,2]
=> [2]
=> 2
[3,2,1,5,4] => [2,1,2] => [2,2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [2,3] => [3,2]
=> [2]
=> 2
[3,2,4,5,1] => [2,3] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [2,3] => [3,2]
=> [2]
=> 2
[3,2,5,4,1] => [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[3,4,1,2,5] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,1,5,2] => [3,2] => [3,2]
=> [2]
=> 2
[3,4,2,1,5] => [3,2] => [3,2]
=> [2]
=> 2
[3,4,2,5,1] => [3,2] => [3,2]
=> [2]
=> 2
[3,4,5,1,2] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,4,5,2,1] => [4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,1,2,4] => [5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[3,5,1,4,2] => [4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
Description
The least common multiple of the parts of the partition.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001060The distinguishing index of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001118The acyclic chromatic index of a graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000937The number of positive values of the symmetric group character corresponding to the partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000993The multiplicity of the largest part of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000456The monochromatic index of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000815The number of semistandard Young tableaux of partition weight of given shape. St001128The exponens consonantiae of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000007The number of saliances of the permutation. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2.
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