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Matching statistic: St000054
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1] => 1
[1,3,2] => [1,2] => [1] => 1
[2,1,3] => [2,1] => [1] => 1
[2,3,1] => [2,1] => [1] => 1
[3,1,2] => [1,2] => [1] => 1
[3,2,1] => [2,1] => [1] => 1
[1,2,3,4] => [1,2,3] => [1,2] => 1
[1,2,4,3] => [1,2,3] => [1,2] => 1
[1,3,2,4] => [1,3,2] => [1,2] => 1
[1,3,4,2] => [1,3,2] => [1,2] => 1
[1,4,2,3] => [1,2,3] => [1,2] => 1
[1,4,3,2] => [1,3,2] => [1,2] => 1
[2,1,3,4] => [2,1,3] => [2,1] => 2
[2,1,4,3] => [2,1,3] => [2,1] => 2
[2,3,1,4] => [2,3,1] => [2,1] => 2
[2,3,4,1] => [2,3,1] => [2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1] => 2
[2,4,3,1] => [2,3,1] => [2,1] => 2
[3,1,2,4] => [3,1,2] => [1,2] => 1
[3,1,4,2] => [3,1,2] => [1,2] => 1
[3,2,1,4] => [3,2,1] => [2,1] => 2
[3,2,4,1] => [3,2,1] => [2,1] => 2
[3,4,1,2] => [3,1,2] => [1,2] => 1
[3,4,2,1] => [3,2,1] => [2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2] => 1
[4,1,3,2] => [1,3,2] => [1,2] => 1
[4,2,1,3] => [2,1,3] => [2,1] => 2
[4,2,3,1] => [2,3,1] => [2,1] => 2
[4,3,1,2] => [3,1,2] => [1,2] => 1
[4,3,2,1] => [3,2,1] => [2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,2] => 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,2] => 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,2] => 1
[1,4,2,3,5] => [1,4,2,3] => [1,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3] => 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,3] => 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
Matching statistic: St000740
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1] => 1
[1,3,2] => [1,2] => [1] => 1
[2,1,3] => [2,1] => [1] => 1
[2,3,1] => [2,1] => [1] => 1
[3,1,2] => [1,2] => [1] => 1
[3,2,1] => [2,1] => [1] => 1
[1,2,3,4] => [1,2,3] => [1,2] => 2
[1,2,4,3] => [1,2,3] => [1,2] => 2
[1,3,2,4] => [1,3,2] => [1,2] => 2
[1,3,4,2] => [1,3,2] => [1,2] => 2
[1,4,2,3] => [1,2,3] => [1,2] => 2
[1,4,3,2] => [1,3,2] => [1,2] => 2
[2,1,3,4] => [2,1,3] => [2,1] => 1
[2,1,4,3] => [2,1,3] => [2,1] => 1
[2,3,1,4] => [2,3,1] => [2,1] => 1
[2,3,4,1] => [2,3,1] => [2,1] => 1
[2,4,1,3] => [2,1,3] => [2,1] => 1
[2,4,3,1] => [2,3,1] => [2,1] => 1
[3,1,2,4] => [3,1,2] => [1,2] => 2
[3,1,4,2] => [3,1,2] => [1,2] => 2
[3,2,1,4] => [3,2,1] => [2,1] => 1
[3,2,4,1] => [3,2,1] => [2,1] => 1
[3,4,1,2] => [3,1,2] => [1,2] => 2
[3,4,2,1] => [3,2,1] => [2,1] => 1
[4,1,2,3] => [1,2,3] => [1,2] => 2
[4,1,3,2] => [1,3,2] => [1,2] => 2
[4,2,1,3] => [2,1,3] => [2,1] => 1
[4,2,3,1] => [2,3,1] => [2,1] => 1
[4,3,1,2] => [3,1,2] => [1,2] => 2
[4,3,2,1] => [3,2,1] => [2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3] => 3
[1,2,3,5,4] => [1,2,3,4] => [1,2,3] => 3
[1,2,4,3,5] => [1,2,4,3] => [1,2,3] => 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,3] => 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3] => 3
[1,2,5,4,3] => [1,2,4,3] => [1,2,3] => 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,2,3] => 3
[1,4,2,5,3] => [1,4,2,3] => [1,2,3] => 3
[1,4,3,2,5] => [1,4,3,2] => [1,3,2] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,3,2] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,2,3] => 3
[1,4,5,3,2] => [1,4,3,2] => [1,3,2] => 2
[1,5,2,3,4] => [1,2,3,4] => [1,2,3] => 3
[1,5,2,4,3] => [1,2,4,3] => [1,2,3] => 3
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001806
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001806: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St001806: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1] => 1
[1,3,2] => [1,2] => [1] => 1
[2,1,3] => [2,1] => [1] => 1
[2,3,1] => [2,1] => [1] => 1
[3,1,2] => [1,2] => [1] => 1
[3,2,1] => [2,1] => [1] => 1
[1,2,3,4] => [1,2,3] => [1,2] => 2
[1,2,4,3] => [1,2,3] => [1,2] => 2
[1,3,2,4] => [1,3,2] => [1,2] => 2
[1,3,4,2] => [1,3,2] => [1,2] => 2
[1,4,2,3] => [1,2,3] => [1,2] => 2
[1,4,3,2] => [1,3,2] => [1,2] => 2
[2,1,3,4] => [2,1,3] => [2,1] => 1
[2,1,4,3] => [2,1,3] => [2,1] => 1
[2,3,1,4] => [2,3,1] => [2,1] => 1
[2,3,4,1] => [2,3,1] => [2,1] => 1
[2,4,1,3] => [2,1,3] => [2,1] => 1
[2,4,3,1] => [2,3,1] => [2,1] => 1
[3,1,2,4] => [3,1,2] => [1,2] => 2
[3,1,4,2] => [3,1,2] => [1,2] => 2
[3,2,1,4] => [3,2,1] => [2,1] => 1
[3,2,4,1] => [3,2,1] => [2,1] => 1
[3,4,1,2] => [3,1,2] => [1,2] => 2
[3,4,2,1] => [3,2,1] => [2,1] => 1
[4,1,2,3] => [1,2,3] => [1,2] => 2
[4,1,3,2] => [1,3,2] => [1,2] => 2
[4,2,1,3] => [2,1,3] => [2,1] => 1
[4,2,3,1] => [2,3,1] => [2,1] => 1
[4,3,1,2] => [3,1,2] => [1,2] => 2
[4,3,2,1] => [3,2,1] => [2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3] => 2
[1,2,3,5,4] => [1,2,3,4] => [1,2,3] => 2
[1,2,4,3,5] => [1,2,4,3] => [1,2,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3] => 2
[1,2,5,4,3] => [1,2,4,3] => [1,2,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2] => 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2] => 3
[1,3,4,2,5] => [1,3,4,2] => [1,3,2] => 3
[1,3,4,5,2] => [1,3,4,2] => [1,3,2] => 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2] => 3
[1,3,5,4,2] => [1,3,4,2] => [1,3,2] => 3
[1,4,2,3,5] => [1,4,2,3] => [1,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => [1,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => [1,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,3,2] => 3
[1,5,2,3,4] => [1,2,3,4] => [1,2,3] => 2
[1,5,2,4,3] => [1,2,4,3] => [1,2,3] => 2
Description
The upper middle entry of a permutation.
This is the entry \sigma(\frac{n+1}{2}) when n is odd, and \sigma(\frac{n}{2}+1) when n is even, where n is the size of the permutation \sigma.
Matching statistic: St001807
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1] => 1
[1,3,2] => [1,2] => [1] => 1
[2,1,3] => [2,1] => [1] => 1
[2,3,1] => [2,1] => [1] => 1
[3,1,2] => [1,2] => [1] => 1
[3,2,1] => [2,1] => [1] => 1
[1,2,3,4] => [1,2,3] => [1,2] => 1
[1,2,4,3] => [1,2,3] => [1,2] => 1
[1,3,2,4] => [1,3,2] => [1,2] => 1
[1,3,4,2] => [1,3,2] => [1,2] => 1
[1,4,2,3] => [1,2,3] => [1,2] => 1
[1,4,3,2] => [1,3,2] => [1,2] => 1
[2,1,3,4] => [2,1,3] => [2,1] => 2
[2,1,4,3] => [2,1,3] => [2,1] => 2
[2,3,1,4] => [2,3,1] => [2,1] => 2
[2,3,4,1] => [2,3,1] => [2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1] => 2
[2,4,3,1] => [2,3,1] => [2,1] => 2
[3,1,2,4] => [3,1,2] => [1,2] => 1
[3,1,4,2] => [3,1,2] => [1,2] => 1
[3,2,1,4] => [3,2,1] => [2,1] => 2
[3,2,4,1] => [3,2,1] => [2,1] => 2
[3,4,1,2] => [3,1,2] => [1,2] => 1
[3,4,2,1] => [3,2,1] => [2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2] => 1
[4,1,3,2] => [1,3,2] => [1,2] => 1
[4,2,1,3] => [2,1,3] => [2,1] => 2
[4,2,3,1] => [2,3,1] => [2,1] => 2
[4,3,1,2] => [3,1,2] => [1,2] => 1
[4,3,2,1] => [3,2,1] => [2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3] => 2
[1,2,3,5,4] => [1,2,3,4] => [1,2,3] => 2
[1,2,4,3,5] => [1,2,4,3] => [1,2,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3] => 2
[1,2,5,4,3] => [1,2,4,3] => [1,2,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2] => 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2] => 3
[1,3,4,2,5] => [1,3,4,2] => [1,3,2] => 3
[1,3,4,5,2] => [1,3,4,2] => [1,3,2] => 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2] => 3
[1,3,5,4,2] => [1,3,4,2] => [1,3,2] => 3
[1,4,2,3,5] => [1,4,2,3] => [1,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => [1,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => [1,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,3,2] => 3
[1,5,2,3,4] => [1,2,3,4] => [1,2,3] => 2
[1,5,2,4,3] => [1,2,4,3] => [1,2,3] => 2
Description
The lower middle entry of a permutation.
This is the entry \sigma(\frac{n+1}{2}) when n is odd, and \sigma(\frac{n}{2}) when n is even, where n is the size of the permutation \sigma.
Matching statistic: St000744
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 3
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 3
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St000460
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1}
[1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1}
[2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1}
[3,1,2] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1}
[3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 3
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 3
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000771
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [3,2,1] => ([],3)
=> ([],3)
=> ? ∊ {1,1,1}
[3,1,2] => [3,2,1] => ([],3)
=> ([],3)
=> ? ∊ {1,1,1}
[3,2,1] => [3,2,1] => ([],3)
=> ([],3)
=> ? ∊ {1,1,1}
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [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] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [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] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,3,1] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,1,2] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,2,1] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,2,3] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,3,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,2,1,3] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,2,3,1] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,1,2] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,2,4,3,5] => [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)
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [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)
=> 1
[1,3,2,5,4] => [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] => [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
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [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
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,2,5] => [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] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,4,3,5] => [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] => [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
[2,1,5,3,4] => [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
[2,1,5,4,3] => [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] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,1,5,4] => [3,2,1,5,4] => ([(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
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[2,3,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[2,4,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,2,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,1,2] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,2,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,2,5,3] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,3,5,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,2,3] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,3,2] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,2,1,5,3] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,2,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,2,5,1,3] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,2,5,3,1] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,3,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,3,2,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,3,5,1,2] => [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
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: St001630
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Values
[1,2,3] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
[1,3,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
[2,1,3] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1}
[2,3,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1}
[3,1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
[3,2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1}
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,4,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000772
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1}
[1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1}
[2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1}
[2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1}
[3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1}
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,4,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,1,3] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,1,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2,6,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The multiplicity of the largest 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 1.
The path on four vertices has eigenvalues 0, 4.7\dots, 6, 9.2\dots and therefore also statistic 1.
The graphs with statistic n-1, n-2 and n-3 have been characterised, see [1].
Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1}
[2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1}
[2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1}
[3,1,2] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1}
[3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 2
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 1
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 1
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> 2
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> 2
[1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 1
[1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 1
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 1
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 1
Description
The multiplicity of the largest part of an integer partition.
The following 169 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St000259The diameter of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001557The number of inversions of the second entry of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000422The energy of a graph, if it is integral. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000717The number of ordinal summands of a poset. St001964The interval resolution global dimension of a poset. St001556The number of inversions of the third entry of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001623The number of doubly irreducible elements of a lattice. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001644The dimension of a graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000022The number of fixed points of a permutation. St000245The number of ascents of a permutation. St000731The number of double exceedences of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000097The order of the largest clique of the graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001272The number of graphs with the same degree sequence. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001638The book thickness of a graph. St001656The monophonic position number of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000370The genus of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000068The number of minimal elements in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000098The chromatic number of a graph. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001946The number of descents in a parking function. St000905The number of different multiplicities of parts of an integer composition. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001171The vector space dimension of Ext_A^1(I_o,A) when I_o is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(x^n). St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001857The number of edges in the reduced word graph of a signed permutation. St001581The achromatic number of a graph. St000632The jump number of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001875The number of simple modules with projective dimension at most 1. St001490The number of connected components of a skew partition. St000322The skewness of a graph. St000528The height of a poset. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by 4. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001301The first Betti number of the order complex associated with the poset. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001642The Prague dimension of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001734The lettericity of a graph. St001827The number of two-component spanning forests of a graph. St001871The number of triconnected components of a graph. St000906The length of the shortest maximal chain in a poset. St001391The disjunction number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001828The Euler characteristic of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St001271The competition number of a graph. St001765The number of connected components of the friends and strangers graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000822The Hadwiger number of the graph. St000299The number of nonisomorphic vertex-induced subtrees. St000741The Colin de Verdière graph invariant.
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