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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000074
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St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
=> 0
[[1,0],[1]]
=> 0
[[2,0],[0]]
=> 0
[[2,0],[1]]
=> 1
[[2,0],[2]]
=> 0
[[1,1],[1]]
=> 0
[[1,0,0],[0,0],[0]]
=> 0
[[1,0,0],[1,0],[0]]
=> 0
[[1,0,0],[1,0],[1]]
=> 0
[[3,0],[0]]
=> 0
[[3,0],[1]]
=> 1
[[3,0],[2]]
=> 1
[[3,0],[3]]
=> 0
[[2,1],[1]]
=> 0
[[2,1],[2]]
=> 0
[[2,0,0],[0,0],[0]]
=> 0
[[2,0,0],[1,0],[0]]
=> 1
[[2,0,0],[1,0],[1]]
=> 1
[[2,0,0],[2,0],[0]]
=> 0
[[2,0,0],[2,0],[1]]
=> 1
[[2,0,0],[2,0],[2]]
=> 0
[[1,1,0],[1,0],[0]]
=> 0
[[1,1,0],[1,0],[1]]
=> 0
[[1,1,0],[1,1],[1]]
=> 0
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> 0
[[4,0],[0]]
=> 0
[[4,0],[1]]
=> 1
[[4,0],[2]]
=> 1
[[4,0],[3]]
=> 1
[[4,0],[4]]
=> 0
[[3,1],[1]]
=> 0
[[3,1],[2]]
=> 1
[[3,1],[3]]
=> 0
[[2,2],[2]]
=> 0
[[3,0,0],[0,0],[0]]
=> 0
[[3,0,0],[1,0],[0]]
=> 1
[[3,0,0],[1,0],[1]]
=> 1
[[3,0,0],[2,0],[0]]
=> 1
[[3,0,0],[2,0],[1]]
=> 2
[[3,0,0],[2,0],[2]]
=> 1
[[3,0,0],[3,0],[0]]
=> 0
[[3,0,0],[3,0],[1]]
=> 1
[[3,0,0],[3,0],[2]]
=> 1
[[3,0,0],[3,0],[3]]
=> 0
[[2,1,0],[1,0],[0]]
=> 0
[[2,1,0],[1,0],[1]]
=> 0
[[2,1,0],[1,1],[1]]
=> 0
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000259
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The Gutman (or modified Schultz) index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 0
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The hyper-Wiener index of a connected graph.
This is
$$
\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.
$$
Matching statistic: St000771
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 + 1
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 + 1
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000777
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 + 1
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001645
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 9%
Values
[[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0],[0]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 0 + 1
[[4,0],[1]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 1 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 2 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 + 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[5,0,0],[5,0],[5]]
=> [[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,1,0,0],[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,3,0],[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The pebbling number of a connected graph.
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