searching the database
Your data matches 74 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000110
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001855
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St001346
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
St001346: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 6
[1,3,2] => 2
[2,1,3] => 3
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 24
[1,2,4,3] => 6
[1,3,2,4] => 8
[1,3,4,2] => 4
[1,4,2,3] => 6
[1,4,3,2] => 2
[2,1,3,4] => 12
[2,1,4,3] => 3
[2,3,1,4] => 8
[2,3,4,1] => 6
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 8
[3,1,4,2] => 4
[3,2,1,4] => 4
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 2
[4,1,2,3] => 6
[4,1,3,2] => 2
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
Description
The number of parking functions that give the same permutation.
A '''parking function''' $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
Matching statistic: St000100
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => ([],1)
=> ? = 1
[1,2] => [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[2,1] => [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 8
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 8
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 6
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of linear extensions of a poset.
Matching statistic: St000777
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 2
[2,1] => [2,1] => [1,2] => ([],2)
=> ? = 1
[1,2,3] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,6}
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {1,2,6}
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3
[3,2,1] => [3,2,1] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,2,6}
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[2,3,1,4] => [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[2,3,4,1] => [2,4,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[3,1,2,4] => [3,1,4,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4
[4,2,3,1] => [4,2,3,1] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3,6,6,6,6,8,8,8,12,24}
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,2] => [2] => ([],2)
=> ([],2)
=> 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [3] => ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,6} - 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,6} - 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4] => [4] => ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,6,6,6,6,8,8,8,12,24} - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
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.
Matching statistic: St001545
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => ([],2)
=> ([],1)
=> ? = 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {1,2,3}
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,2,3}
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,2,3}
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 6
[1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,3,3,3,6,8,8,8,12}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 6
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 6
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 6
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 24
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001645
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,2] => ([],2)
=> ([],1)
=> 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,6}
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,6}
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,12,24}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,6} - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,6} - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[2,4,3,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[3,1,4,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[4,2,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,3,3,6,6,6,6,8,8,8,12,24} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000455
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 39%●distinct values known / distinct values provided: 38%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 39%●distinct values known / distinct values provided: 38%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,2] => [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,2,3] => [1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,6} - 2
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,6} - 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,6} - 2
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[2,1,3,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[2,3,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,1,2,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24} - 2
[4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
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.
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St001060The distinguishing index of a graph. St000438The position of the last up step in a Dyck path. St000981The length of the longest zigzag subpath. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St000259The diameter of a connected graph. St001626The number of maximal proper sublattices of a lattice. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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 $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000177The number of free tiles in the pattern. St000178Number of free entries. St000422The energy of a graph, if it is integral. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!