Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001520
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
St001520: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,3,2] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,4,5,2] => 0
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [2,1,5,4,3] => 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,2,4] => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,2,5,4] => 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4,2,5,3] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,3,4,5,2] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,5,2,1,3] => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,4,5,3,2] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1
Description
The number of strict 3-descents. A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Mp00139: Ordered trees Zeilberger's Strahler bijectionBinary trees
Mp00008: Binary trees to complete treeOrdered trees
Mp00050: Ordered trees to binary tree: right brother = right childBinary trees
St000252: Binary trees ⟶ ℤResult quality: 36% values known / values provided: 36%distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [[],[]]
=> [.,[.,.]]
=> 0
[[],[]]
=> [.,[.,.]]
=> [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> 0
[[[]]]
=> [[.,.],.]
=> [[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> 0
[[],[],[]]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> 0
[[],[[]]]
=> [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> 0
[[[],[]]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> 0
[[[[]]]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> ? = 0
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> ? = 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ? = 0
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> ? = 0
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> ? = 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 0
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> ? = 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> ? = 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> ? = 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> ? = 0
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> ? = 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> ? = 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> ? = 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> ? = 1
Description
The number of nodes of degree 3 of a binary tree. Equivalently, the number of internal triangles in the associated triangulation of an $(n+2)$-gon.
Matching statistic: St000365
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000365: Permutations ⟶ ℤResult quality: 36% values known / values provided: 36%distinct values known / distinct values provided: 67%
Values
[[]]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[[[]]]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? = 0
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 1
Description
The number of double ascents of a permutation. A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Mp00139: Ordered trees Zeilberger's Strahler bijectionBinary trees
Mp00008: Binary trees to complete treeOrdered trees
Mp00051: Ordered trees to Dyck pathDyck paths
St001594: Dyck paths ⟶ ℤResult quality: 36% values known / values provided: 36%distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [[],[]]
=> [1,0,1,0]
=> 0
[[],[]]
=> [.,[.,.]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 0
[[[]]]
=> [[.,.],.]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 0
[[],[],[]]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
[[],[[]]]
=> [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0
[[[],[]]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[[[[]]]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 0
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 0
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
Description
The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. See the link for the definition.
Matching statistic: St001964
Mp00046: Ordered trees to graphGraphs
Mp00243: Graphs weak duplicate orderPosets
St001964: Posets ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 33%
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([],2)
=> 0
[[[]]]
=> ([(0,2),(1,2)],3)
=> ([],2)
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0
[[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0
[[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0
[[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0
[[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ? = 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0
[[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ? = 1
[[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1
[[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ? = 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0
[[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2
[[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ? = 1
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Mp00046: Ordered trees to graphGraphs
Mp00247: Graphs de-duplicateGraphs
St000454: Graphs ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 33%
Values
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 1
[[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 1
[[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 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.
Mp00046: Ordered trees to graphGraphs
Mp00247: Graphs de-duplicateGraphs
St000422: Graphs ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 33%
Values
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[[]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Mp00046: Ordered trees to graphGraphs
Mp00247: Graphs de-duplicateGraphs
St000718: Graphs ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 33%
Values
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[[]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
Description
The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]
Mp00046: Ordered trees to graphGraphs
Mp00157: Graphs connected complementGraphs
Mp00259: Graphs vertex additionGraphs
St000455: Graphs ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 33%
Values
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[[[]]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0
[[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St001435
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00233: Dyck paths skew partitionSkew partitions
St001435: Skew partitions ⟶ ℤResult quality: 32% values known / values provided: 32%distinct values known / distinct values provided: 67%
Values
[[]]
=> [1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? = 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? = 0
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? = 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? = 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? = 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? = 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 1
Description
The number of missing boxes in the first row.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001438The number of missing boxes of a skew partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001487The number of inner corners of a skew partition. St000264The girth of a graph, which is not a tree. St001644The dimension of a graph. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001828The Euler characteristic of a graph. St001890The maximum magnitude of the Möbius function of a poset. St000822The Hadwiger number of the graph. St001060The distinguishing index of a graph. St001626The number of maximal proper sublattices of a lattice. St001734The lettericity of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001645The pebbling number of a connected graph.