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Your data matches 101 different statistics following compositions of up to 3 maps.
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Matching statistic: St000481
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 0
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 0
[2,1,3] => ([(1,2)],3)
=> [1]
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> [2]
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [2]
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [2]
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [2]
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [2]
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [2]
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [2]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [2]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [2]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [2]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000183
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000480
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> []
=> 0
[1,2] => ([],2)
=> [1,1]
=> [1]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> []
=> 0
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 0
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> 0
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 0
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000660
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1,0]
=> 0
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000920
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001597
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [[2,2],[]]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[2,2,1],[]]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1 = 0 + 1
Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000379
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0
[1,2] => [2,1] => ([(0,1)],2)
=> ([],1)
=> ? = 0
[2,1] => [1,2] => ([],2)
=> ([],2)
=> 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? = 0
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> ([],2)
=> 0
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> ([],2)
=> 0
[3,2,1] => [1,2,3] => ([],3)
=> ([],3)
=> 0
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 1
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> 0
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> 0
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> 0
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> 0
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,5,3,2,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,5,4,2,3] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
[2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[2,4,3,1,5] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[2,4,5,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,4,3,5,6] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
[1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,4,6,5,3] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000456
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[2,1] => ([(0,1)],2)
=> ([],1)
=> ? = 0 + 1
[1,2,3] => ([],3)
=> ([],3)
=> ? = 0 + 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? = 0 + 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 1
[1,2,3,4] => ([],4)
=> ([],4)
=> ? = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ? = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ? = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ? = 0 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ? = 0 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0
[1,2] => [2,1] => [2] => ([],2)
=> ? = 0
[2,1] => [1,2] => [2] => ([],2)
=> ? = 0
[1,2,3] => [2,3,1] => [3] => ([],3)
=> ? = 0
[1,3,2] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 0
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? = 0
[3,1,2] => [3,1,2] => [3] => ([],3)
=> ? = 0
[3,2,1] => [2,1,3] => [3] => ([],3)
=> ? = 0
[1,2,3,4] => [2,3,4,1] => [4] => ([],4)
=> ? = 0
[1,2,4,3] => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,1,3,4] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 0
[2,1,4,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[2,3,1,4] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[2,4,1,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 0
[2,4,3,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 0
[3,1,2,4] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,1,4] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [2,1,3,4] => [4] => ([],4)
=> ? = 0
[3,4,1,2] => [4,1,2,3] => [4] => ([],4)
=> ? = 0
[3,4,2,1] => [3,1,2,4] => [4] => ([],4)
=> ? = 0
[4,1,2,3] => [3,4,1,2] => [4] => ([],4)
=> ? = 0
[4,1,3,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 0
[4,2,1,3] => [2,4,1,3] => [4] => ([],4)
=> ? = 0
[4,2,3,1] => [2,3,1,4] => [4] => ([],4)
=> ? = 0
[4,3,1,2] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[4,3,2,1] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => [2,3,4,5,1] => [5] => ([],5)
=> ? = 0
[1,2,3,5,4] => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3,5] => [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,5,3] => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,3,2,4,5] => [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,5,4] => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,3,4,2,5] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,3,5,4,2] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,4,2,3,5] => [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,4,3,2,5] => [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,4,3,5,2] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,4,5,2,3] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,4,5,3,2] => [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,2,3,4] => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,3,2,4] => [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,3,4,2] => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,4,2,3] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [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)
=> ? = 0
[2,1,3,4,5] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 0
[2,1,3,5,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,4,3,5] => [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,4,5,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,5,3,4] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,5,4,3] => [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
[2,3,1,4,5] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,1,5,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,3,4,1,5] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0
[2,3,5,1,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,5,4,1] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,4,1,5,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,4,3,1,5] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,4,3,5,1] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 0
[2,4,5,1,3] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> 0
[2,4,5,3,1] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 0
[2,5,1,3,4] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 0
[2,5,1,4,3] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[2,5,3,1,4] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 0
[2,5,3,4,1] => [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 0
[2,5,4,1,3] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,5,4,3,1] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,1,2,4,5] => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,2,5,4] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,1,4,2,5] => [4,1,3,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,1,4,5,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[3,1,5,2,4] => [4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,1,5,4,2] => [5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,2,1,4,5] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2,1,5,4] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,2,4,1,5] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,2,4,5,1] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 0
[3,2,5,1,4] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2,5,4,1] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,4,1,2,5] => [4,1,2,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,4,1,5,2] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[3,4,2,1,5] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,4,2,5,1] => [3,1,2,4,5] => [5] => ([],5)
=> ? = 0
[3,4,5,1,2] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 0
[3,4,5,2,1] => [4,1,2,3,5] => [5] => ([],5)
=> ? = 0
[4,1,3,2,5] => [4,3,1,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001964
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[1] => => ?
=> ? = 0
[1,2] => 1 => ([(0,1)],2)
=> 0
[2,1] => 0 => ([(0,1)],2)
=> 0
[1,2,3] => 11 => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,1] => 00 => ([(0,2),(2,1)],3)
=> 0
[3,1,2] => 00 => ([(0,2),(2,1)],3)
=> 0
[3,2,1] => 00 => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[1,3,2,4] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0
[1,3,4,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[1,4,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[1,4,3,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,3,4] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,4,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
[2,3,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,3,4,1] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,3,1] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[3,1,4,2] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,2,1,4] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[3,2,4,1] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,1,2] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,3,2] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,3,1] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,1,2] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,2,1] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,4,3,5] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[1,2,4,5,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[1,2,5,3,4] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[1,2,5,4,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[1,3,2,4,5] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[1,3,2,5,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[1,3,4,2,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0
[1,3,4,5,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,3,5,2,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,3,5,4,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,4,2,3,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0
[1,4,2,5,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,4,3,2,5] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0
[1,4,3,5,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,4,5,2,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,4,5,3,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,5,2,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,5,2,4,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,5,3,2,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,5,3,4,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,5,4,2,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,5,4,3,2] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,1,3,4,5] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,1,3,5,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[2,1,4,3,5] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[2,1,4,5,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[2,1,5,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[2,1,5,4,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[2,3,1,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[2,3,1,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[2,3,4,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,1,3,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,4,1,5,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,3,1,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,4,3,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,5,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,5,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,3,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,5,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,1,2,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[3,1,2,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[3,1,4,2,5] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[3,1,4,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,1,5,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,1,5,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,1,4,5] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[3,2,1,5,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[3,2,4,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,5,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,5,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,1,5,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,2,5,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,5,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,5,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,1,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,1,4,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,2,1,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001490The number of connected components of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001845The number of join irreducibles minus the rank of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of 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. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs 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. St001325The minimal number of occurrences of the comparability-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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the 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. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001867The number of alignments of type EN of a signed permutation. St000322The skewness of a graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. 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. St000449The number of pairs of vertices of a graph with distance 4. St001396Number of triples of incomparable elements in a finite 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. St001578The minimal number of edges to add or remove to make a graph a line 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. St001871The number of triconnected components of a graph. 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. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000264The girth of a graph, which is not a tree.
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