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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000739
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [[1,2]]
=> [[1,2]]
=> [[1,2]]
=> 1
[[2,2]]
=> [[2,2]]
=> [[2,2]]
=> [[2,2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> [[1,2]]
=> [[1,2]]
=> 1
[[1,1,2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> 1
[[1,2,2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> 1
[[2,2,2]]
=> [[2,2,2]]
=> [[2,2,2]]
=> [[2,2,2]]
=> 2
[[1,1],[2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> [[1,1,2]]
=> 1
[[1,2],[2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> [[1,2,2]]
=> 1
[[1,1,3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> 1
[[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3,3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> 1
[[2,2,3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> 2
[[2,3,3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> 2
[[3,3,3]]
=> [[3,3,3]]
=> [[3,3,3]]
=> [[3,3,3]]
=> 3
[[1,1],[3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> [[1,1,3]]
=> 1
[[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3],[2]]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> [[1,3,3]]
=> 1
[[2,2],[3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> [[2,2,3]]
=> 2
[[2,3],[3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> [[2,3,3]]
=> 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [[2,2,2,2]]
=> [[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> [[1,2,2,2]]
=> 1
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> 1
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 1
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> 1
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 1
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> 1
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> 1
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> 2
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> 2
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [[2,3,3,3]]
=> [[2,3,3,3]]
=> 2
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [[3,3,3,3]]
=> [[3,3,3,3]]
=> 3
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> [[1,1,1,3]]
=> 1
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [[1,1,2,3]]
=> [[1,1,2,3]]
=> 1
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> [[1,1,3,3]]
=> 1
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 1
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [[1,2,2,3]]
=> [[1,2,2,3]]
=> 1
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> 1
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [[1,2,3,3]]
=> [[1,2,3,3]]
=> 1
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> [[1,3,3,3]]
=> 1
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> [[2,2,2,3]]
=> 2
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> [[2,2,3,3]]
=> 2
Description
The first entry in the last row of a semistandard tableau.
Matching statistic: St001410
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(load all 4 compositions to match this statistic)
St001410: Semistandard tableaux ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 80%
Values
[[1,2]]
=> 1
[[2,2]]
=> 2
[[1],[2]]
=> 1
[[1,1,2]]
=> 1
[[1,2,2]]
=> 1
[[2,2,2]]
=> 2
[[1,1],[2]]
=> 1
[[1,2],[2]]
=> 1
[[1,1,3]]
=> 1
[[1,2,3]]
=> 1
[[1,3,3]]
=> 1
[[2,2,3]]
=> 2
[[2,3,3]]
=> 2
[[3,3,3]]
=> 3
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 1
[[1,3],[2]]
=> 1
[[1,3],[3]]
=> 1
[[2,2],[3]]
=> 2
[[2,3],[3]]
=> 2
[[1],[2],[3]]
=> 1
[[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> 1
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 1
[[1,1,1,3]]
=> 1
[[1,1,2,3]]
=> 1
[[1,1,3,3]]
=> 1
[[1,2,2,3]]
=> 1
[[1,2,3,3]]
=> 1
[[1,3,3,3]]
=> 1
[[2,2,2,3]]
=> 2
[[2,2,3,3]]
=> 2
[[2,3,3,3]]
=> 2
[[3,3,3,3]]
=> 3
[[1,1,1],[3]]
=> 1
[[1,1,2],[3]]
=> 1
[[1,1,3],[2]]
=> 1
[[1,1,3],[3]]
=> 1
[[1,2,2],[3]]
=> 1
[[1,2,3],[2]]
=> 1
[[1,2,3],[3]]
=> 1
[[1,3,3],[2]]
=> 1
[[1,3,3],[3]]
=> 1
[[2,2,2],[3]]
=> 2
[[2,2,3],[3]]
=> 2
[[1,1,1,1,4]]
=> ? = 1
[[1,1,1,2,4]]
=> ? = 1
[[1,1,1,3,4]]
=> ? = 1
[[1,1,1,4,4]]
=> ? = 1
[[1,1,2,2,4]]
=> ? = 1
[[1,1,2,3,4]]
=> ? = 1
[[1,1,2,4,4]]
=> ? = 1
[[1,1,3,3,4]]
=> ? = 1
[[1,1,3,4,4]]
=> ? = 1
[[1,1,4,4,4]]
=> ? = 1
[[1,2,2,2,4]]
=> ? = 1
[[1,2,2,3,4]]
=> ? = 1
[[1,2,2,4,4]]
=> ? = 1
[[1,2,3,3,4]]
=> ? = 1
[[1,2,3,4,4]]
=> ? = 1
[[1,2,4,4,4]]
=> ? = 1
[[1,3,3,3,4]]
=> ? = 1
[[1,3,3,4,4]]
=> ? = 1
[[1,3,4,4,4]]
=> ? = 1
[[1,4,4,4,4]]
=> ? = 1
[[2,2,2,2,4]]
=> ? = 2
[[2,2,2,3,4]]
=> ? = 2
[[2,2,2,4,4]]
=> ? = 2
[[2,2,3,3,4]]
=> ? = 2
[[2,2,3,4,4]]
=> ? = 2
[[2,2,4,4,4]]
=> ? = 2
[[2,3,3,3,4]]
=> ? = 2
[[2,3,3,4,4]]
=> ? = 2
[[2,3,4,4,4]]
=> ? = 2
[[2,4,4,4,4]]
=> ? = 2
[[3,3,3,3,4]]
=> ? = 3
[[3,3,3,4,4]]
=> ? = 3
[[3,3,4,4,4]]
=> ? = 3
[[3,4,4,4,4]]
=> ? = 3
[[4,4,4,4,4]]
=> ? = 4
[[1,1,1,1],[4]]
=> ? = 1
[[1,1,1,2],[4]]
=> ? = 1
[[1,1,1,4],[2]]
=> ? = 1
[[1,1,1,3],[4]]
=> ? = 1
[[1,1,1,4],[3]]
=> ? = 1
[[1,1,1,4],[4]]
=> ? = 1
[[1,1,2,2],[4]]
=> ? = 1
[[1,1,2,4],[2]]
=> ? = 1
[[1,1,2,3],[4]]
=> ? = 1
[[1,1,2,4],[3]]
=> ? = 1
[[1,1,3,4],[2]]
=> ? = 1
[[1,1,2,4],[4]]
=> ? = 1
[[1,1,4,4],[2]]
=> ? = 1
[[1,1,3,3],[4]]
=> ? = 1
[[1,1,3,4],[3]]
=> ? = 1
Description
The minimal entry of a semistandard tableau.
Matching statistic: St000455
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[2,2]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 2 - 1
[[1],[2]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[1,1,2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[1,2,2]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[2,2,2]]
=> [[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 2 - 1
[[1,1],[2]]
=> [[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[1,2],[2]]
=> [[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,1,3]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,2,3]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,3,3]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[2,2,3]]
=> [[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)
=> ? = 2 - 1
[[2,3,3]]
=> [[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)
=> ? = 2 - 1
[[3,3,3]]
=> [[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)
=> ? = 3 - 1
[[1,1],[3]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,2],[3]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,3],[2]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,3],[3]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[2,2],[3]]
=> [[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)
=> ? = 2 - 1
[[2,3],[3]]
=> [[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)
=> ? = 2 - 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[1,1,2,2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,2,2,2]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 1
[[2,2,2,2]]
=> [[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 2 - 1
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 1
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,1,1,3]]
=> [[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,1,2,3]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,3,3]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,2,2,3]]
=> [[1,2,2,3]]
=> ([(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
[[1,2,3,3]]
=> [[1,2,3,3]]
=> ([(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
[[1,3,3,3]]
=> [[1,3,3,3]]
=> ([(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)
=> ? = 1 - 1
[[2,2,2,3]]
=> [[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[[2,2,3,3]]
=> [[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
[[2,3,3,3]]
=> [[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 2 - 1
[[3,3,3,3]]
=> [[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? = 3 - 1
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> ([(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
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 1
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> ([(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
[[1,3,3],[2]]
=> [[1,2,3],[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)
=> ? = 1 - 1
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> ([(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)
=> ? = 1 - 1
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
[[2,3,3],[3]]
=> [[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? = 2 - 1
[[1,1],[2,3]]
=> [[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [[1,2,2,3]]
=> ([(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
[[1,2],[3,3]]
=> [[1,2,3,3]]
=> ([(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
[[2,2],[3,3]]
=> [[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 - 1
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 1
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 1
[[1,3],[2],[3]]
=> [[1,2,3],[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)
=> ? = 1 - 1
[[1,1,1,1,2]]
=> [[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,2,3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,3,3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,2],[3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,3],[3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1],[2,3]]
=> [[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,4],[2]]
=> [[1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,4],[3]]
=> [[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1],[2],[4]]
=> [[1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1],[3],[4]]
=> [[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,1,2,3]]
=> [[1,1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,1,3,3]]
=> [[1,1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,1,2],[3]]
=> [[1,1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,1,3],[3]]
=> [[1,1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,1],[2,3]]
=> [[1,1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,1],[3,3]]
=> [[1,1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,1,4],[2]]
=> [[1,1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,1,4],[3]]
=> [[1,1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 1
[[1,1,1,1],[2],[4]]
=> [[1,1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 0 = 1 - 1
[[1,1,1,1],[3],[4]]
=> [[1,1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 0 = 1 - 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.
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