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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000644
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 2
[1,1,1]
=> 0
[4]
=> 4
[3,1]
=> 2
[2,2]
=> 3
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 3
[4,1]
=> 8
[3,2]
=> 8
[3,1,1]
=> 6
[2,2,1]
=> 7
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 0
[6]
=> 8
[5,1]
=> 8
[4,2]
=> 32
[4,1,1]
=> 18
[3,3]
=> 4
[3,2,1]
=> 34
[3,1,1,1]
=> 20
[2,2,2]
=> 16
[2,2,1,1]
=> 14
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 0
[7]
=> 6
[6,1]
=> 30
[5,2]
=> 40
[5,1,1]
=> 60
[4,3]
=> 42
[4,2,1]
=> 184
[4,1,1,1]
=> 82
[3,3,1]
=> 80
[3,2,2]
=> 104
[3,2,1,1]
=> 246
[3,1,1,1,1]
=> 30
[2,2,2,1]
=> 84
[2,2,1,1,1]
=> 54
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 0
[8]
=> 22
[7,1]
=> 44
[6,2]
=> 238
[6,1,1]
=> 210
[5,3]
=> 66
[5,2,1]
=> 588
Description
The number of graphs with given frequency partition.
The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur.
For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$.
There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.
Matching statistic: St000777
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 0
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {0,2}
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {0,2}
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,2,3,4}
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {0,2,3,4}
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4}
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4}
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,3,6,7,8,8}
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34}
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001200
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 11%●distinct values known / distinct values provided: 7%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 11%●distinct values known / distinct values provided: 7%
Values
[1]
=> [1,0]
=> [2,1] => [1,1,0,0]
=> ? = 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,2}
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,2}
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,4}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,4}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,4}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,6,7,8,8}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,6,7,8,8}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,6,7,8,8}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,6,7,8,8}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,6,7,8,8}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,8,8,14,16,18,20,32,34}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001405
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 11%●distinct values known / distinct values provided: 9%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 11%●distinct values known / distinct values provided: 9%
Values
[1]
=> [1,0]
=> [[1],[2]]
=> [2,1] => 1
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? ∊ {0,2,2,3}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? ∊ {0,2,2,3}
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => ? ∊ {0,2,2,3}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => ? ∊ {0,2,2,3}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => ? ∊ {0,2,3,6,7,8,8}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => ? ∊ {0,2,3,6,7,8,8}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ? ∊ {0,2,3,6,7,8,8}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,4,7,9,10,1,3,5,6,8] => ? ∊ {0,2,3,6,7,8,8}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => ? ∊ {0,2,3,6,7,8,8}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => ? ∊ {0,2,3,6,7,8,8}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => ? ∊ {0,2,3,6,7,8,8}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [2,4,6,8,11,12,1,3,5,7,9,10] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,4,8,9,10,1,3,5,6,7] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> [2,4,6,9,11,12,1,3,5,7,8,10] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [2,4,7,9,11,12,1,3,5,6,8,10] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [4,5,7,9,10,1,2,3,6,8] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> [2,5,7,9,11,12,1,3,4,6,8,10] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> [3,5,7,9,11,12,1,2,4,6,8,10] => ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> [2,4,6,8,10,12,14,1,3,5,7,9,11,13] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> [2,4,6,8,10,13,14,1,3,5,7,9,11,12] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,4,6,10,11,12,1,3,5,7,8,9] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,9,10,12],[2,4,6,8,11,13,14]]
=> [2,4,6,8,11,13,14,1,3,5,7,9,10,12] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,6,8,9,10,1,3,4,5,7] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,6,7,10],[2,4,8,9,11,12]]
=> [2,4,8,9,11,12,1,3,5,6,7,10] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> [2,4,6,9,11,13,14,1,3,5,7,8,10,12] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [4,6,7,9,10,1,2,3,5,8] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,7,8,9,10,1,3,4,5,6] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> [2,6,7,9,11,12,1,3,4,5,8,10] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10,12],[2,4,7,9,11,13,14]]
=> [2,4,7,9,11,13,14,1,3,5,6,8,10,12] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [5,6,7,9,10,1,2,3,4,8] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10],[4,5,7,9,11,12]]
=> [4,5,7,9,11,12,1,2,3,6,8,10] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
=> [2,5,7,9,11,13,14,1,3,4,6,8,10,12] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> [3,5,7,9,11,13,14,1,2,4,6,8,10,12] => ? ∊ {0,2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> [2,4,6,8,10,12,14,16,1,3,5,7,9,11,13,15] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> [2,4,6,8,10,12,15,16,1,3,5,7,9,11,13,14] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> [2,4,6,8,12,13,14,1,3,5,7,9,10,11] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,9,11,12,14],[2,4,6,8,10,13,15,16]]
=> [2,4,6,8,10,13,15,16,1,3,5,7,9,11,12,14] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,4,8,10,11,12,1,3,5,6,7,9] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,7,8,9,12],[2,4,6,10,11,13,14]]
=> [2,4,6,10,11,13,14,1,3,5,7,8,9,12] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,9,10,12,14],[2,4,6,8,11,13,15,16]]
=> [2,4,6,8,11,13,15,16,1,3,5,7,9,10,12,14] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [4,6,8,9,10,1,2,3,5,7] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> [2,6,8,9,11,12,1,3,4,5,7,10] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,6,7,8],[2,4,9,10,11,12]]
=> [2,4,9,10,11,12,1,3,5,6,7,8] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,5,6,7,10,12],[2,4,8,9,11,13,14]]
=> [2,4,8,9,11,13,14,1,3,5,6,7,10,12] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12,14],[2,4,6,9,11,13,15,16]]
=> [2,4,6,9,11,13,15,16,1,3,5,7,8,10,12,14] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [4,7,8,9,10,1,2,3,5,6] => ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
Description
The number of bonds in a permutation.
For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St001491
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Values
[1]
=> []
=> []
=> => ? = 1
[2]
=> []
=> []
=> => ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[3]
=> []
=> []
=> => ? ∊ {2,2}
[2,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? ∊ {2,2}
[4]
=> []
=> []
=> => ? ∊ {2,2,3,4}
[3,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? ∊ {2,2,3,4}
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? ∊ {2,2,3,4}
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {2,2,3,4}
[5]
=> []
=> []
=> => ? ∊ {2,3,6,7,8,8}
[4,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? ∊ {2,3,6,7,8,8}
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? ∊ {2,3,6,7,8,8}
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? ∊ {2,3,6,7,8,8}
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {2,3,6,7,8,8}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {2,3,6,7,8,8}
[6]
=> []
=> []
=> => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[5,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {2,4,8,8,14,16,18,20,32,34}
[7]
=> []
=> []
=> => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[6,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? ∊ {2,6,30,30,40,42,54,60,80,82,84,104,184,246}
[8]
=> []
=> []
=> => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[7,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? ∊ {2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 0 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {0,2} - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {0,2} - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,2,3,4} - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {0,2,3,4} - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4} - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4} - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 0 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {0,2} - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {0,2} - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,2,3,4} - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {0,2,3,4} - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4} - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4} - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8} - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,4,8,8,14,16,18,20,32,34} - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246} - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,2,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756} - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000454
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 1
[2]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[1,1]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[3]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,2,2}
[2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,2,2}
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,2,2}
[4]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,3,4}
[2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,3,4}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,3,4}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,3,6,7,8,8}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,3,6,7,8,8}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,3,6,7,8,8}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,3,6,7,8,8}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,3,6,7,8,8}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,2,4,8,8,14,16,18,20,32,34}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[[[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[],[[[]],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,6,30,30,40,42,54,60,80,82,84,104,184,246}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[],[[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[],[],[[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[],[],[[[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[],[],[[],[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[],[],[[],[],[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? ∊ {0,22,30,44,66,160,210,210,238,342,374,426,588,622,630,766,1134,1398,1588,1740,1756}
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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