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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
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(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => ([],1)
=> 0
[2,1] => 0 => [1] => ([],1)
=> 0
[1,3,2] => 10 => [1,1] => ([(0,1)],2)
=> 1
[2,1,3] => 01 => [1,1] => ([(0,1)],2)
=> 1
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,3,5,4,6] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,6,5] => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4,6] => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001418
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 100%●distinct values known / distinct values provided: 67%
Values
[1,2] => 1 => [1] => [1,0]
=> ? = 0 - 1
[2,1] => 0 => [1] => [1,0]
=> ? = 0 - 1
[1,3,2] => 10 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,3] => 01 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3,5,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,4,3,5] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,3,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,3,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,1,2,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,4,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,2,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,2,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,4,1,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,4,2,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,1,2,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,1,3,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,3,1,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,3,2,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,3,4,6,5] => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,3,5,4,6] => 11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3,6] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4,6] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,4,3,6] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,6,5] => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4,6] => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,6,5] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,4,2,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,6,5] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2,6,5] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,3,2,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St001093
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => ([],1)
=> 1 = 0 + 1
[2,1] => 0 => [1] => ([],1)
=> 1 = 0 + 1
[1,3,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,3,5,4,6] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,6,5] => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,5,4,6] => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,2,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,6,5,4,3,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,5,6,4,3,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,5,4,6,3,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,4,5,6,3,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,3,2,4,5,7,6,8] => 1011101 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,6,4,5,3,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,5,3,6,4,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,5,6,3,4,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,6,3,4,5,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,6,3,5,4,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[2,1,6,4,3,5,8,7] => 0100010 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
Description
The detour number of a graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
Matching statistic: St000781
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2]
=> []
=> ? = 0 - 1
[2,1] => [1,2] => [2]
=> []
=> ? = 0 - 1
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 1 - 1
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 1 - 1
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 1
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 1
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 1
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 1
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,1,2,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,1,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,2,1,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,2,4,1,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,4,1,2,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,4,2,1,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,1,3,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,2,3,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,3,1,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,3,2,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,2,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,3,4,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,4,2,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,4,3,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,1,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,1,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,5,1,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,3,5,4,1,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,1,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,1,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,1,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,5,1,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,5,1,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,5,3,1,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,1,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,5,1,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,3,1,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,5,3,4,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,4,1,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,4,3,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 1 - 1
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,2,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[2,1,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[2,1,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2]
=> []
=> ? = 0 - 1
[2,1] => [1,2] => [2]
=> []
=> ? = 0 - 1
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 1 - 1
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 1 - 1
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 1
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 1
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 1
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 1
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,1,2,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,1,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,2,1,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,2,4,1,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,4,1,2,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[3,4,2,1,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,1,3,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,2,3,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,3,1,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[4,3,2,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,2,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,3,4,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,4,2,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,5,4,3,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,1,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,1,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 1
[2,3,5,1,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,3,5,4,1,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,1,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,1,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,1,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,5,1,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,5,1,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,4,5,3,1,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,1,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,5,1,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,3,1,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,5,3,4,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,4,1,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[2,5,4,3,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 1 - 1
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,2,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[2,1,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
[2,1,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St000205
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2]
=> []
=> ? = 0 - 2
[2,1] => [1,2] => [2]
=> []
=> ? = 0 - 2
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 1 - 2
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 1 - 2
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 2
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 2
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 2
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 2
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0 = 2 - 2
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 2
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 2
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[2,4,1,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,1,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,1,2,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,1,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,2,1,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,2,4,1,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,4,1,2,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,4,2,1,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,1,3,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,2,3,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,3,1,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,3,2,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 2
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[1,5,2,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[1,5,3,4,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,4,2,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,4,3,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,1,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 2
[2,1,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,1,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,1,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,3,1,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,1,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,4,1,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,4,5,1,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,5,1,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,3,5,4,1,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,1,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,1,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,1,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,5,1,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,5,1,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,5,3,1,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,1,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,5,1,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,3,1,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,5,3,4,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,4,1,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,4,3,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 1 - 2
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,2,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[2,1,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[2,1,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 78%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2]
=> []
=> ? = 0 - 2
[2,1] => [1,2] => [2]
=> []
=> ? = 0 - 2
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 1 - 2
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 1 - 2
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 2
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 2
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 1 - 2
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 2
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0 = 2 - 2
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 2
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1 - 2
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 2
[2,4,1,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,1,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,1,2,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,1,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,2,1,5,4] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,2,4,1,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,4,1,2,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[3,4,2,1,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,1,3,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,2,3,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,3,1,2,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[4,3,2,1,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 2
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[1,5,2,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[1,5,3,4,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,4,2,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,5,4,3,2,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,1,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 1 - 2
[2,1,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,1,5,3,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,1,5,4,3,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,3,1,4,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,1,5,4,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,4,1,6,5] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,4,5,1,6] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 2 - 2
[2,3,5,1,4,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,3,5,4,1,6] => [1,2,3,5,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,1,3,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,1,5,3,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,1,6,5] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,5,1,6] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,5,1,3,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,4,5,3,1,6] => [1,2,4,3,5,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,1,3,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,5,1,4,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,3,1,4,6] => [1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,5,3,4,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,4,1,3,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[2,5,4,3,1,6] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 1 - 2
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,2,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[2,1,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
[2,1,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 2 - 2
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000455
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Values
[1,2] => 1 => [1] => ([],1)
=> ? = 0 - 2
[2,1] => 0 => [1] => ([],1)
=> ? = 0 - 2
[1,3,2] => 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,1,3] => 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,3,5,4,6] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,3,2,4,6,5] => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,3,2,5,4,6] => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
[1,3,4,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,3,4,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,3,5,2,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,3,5,4,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,2,3,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,2,5,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,3,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,3,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,5,2,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,5,3,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,2,3,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,2,4,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,3,2,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,3,4,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,4,2,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,4,3,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,3,4,6,5] => 01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,3,5,4,6] => 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,4,3,6,5] => 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
[2,1,4,5,3,6] => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,5,3,4,6] => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,1,5,4,3,6] => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,3,1,4,6,5] => 00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,3,1,5,4,6] => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,4,1,6,5] => 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,4,5,1,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,5,1,4,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,5,4,1,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,1,3,6,5] => 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,1,5,3,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,3,1,6,5] => 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,3,5,1,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,5,1,3,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,5,3,1,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,1,3,4,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,1,4,3,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,3,1,4,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,3,4,1,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,4,1,3,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,4,3,1,6] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1,2,4,6,5] => 00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,1,4,6,5] => 00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,2,3,5,6,4,7] => 111001 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,3,6,4,5,7] => 111001 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,3,6,5,4,7] => 111001 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,4,3,5,7,6] => 110110 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,4,5,3,7,6] => 110010 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,4,5,6,3,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,4,6,3,5,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,4,6,5,3,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,5,3,4,7,6] => 110010 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,5,3,6,4,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,5,4,3,7,6] => 110010 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,5,4,6,3,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,5,6,3,4,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,5,6,4,3,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,6,3,4,5,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,2,6,3,5,4,7] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001934
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%
Values
[1,2] => [2]
=> []
=> ?
=> ? = 0 - 1
[2,1] => [1,1]
=> [1]
=> []
=> ? = 0 - 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1 - 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1 - 1
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 2 - 1
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1 - 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1 - 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 2 - 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2 - 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 2 - 1
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 1
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 - 1
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 1
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 2 - 1
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1 - 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 2 - 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2 - 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 2 - 1
[3,1,4,2,5] => [3,2]
=> [2]
=> []
=> ? = 2 - 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,4,1,2,5] => [3,2]
=> [2]
=> []
=> ? = 2 - 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,1,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,2,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,3,1,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,3,4,6,5] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[1,2,3,5,4,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[1,2,4,3,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,2,4,5,3,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[1,2,5,3,4,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,2,4,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,3,2,5,4,6] => [4,2]
=> [2]
=> []
=> ? = 1 - 1
[1,3,4,2,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,3,4,5,2,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[1,3,5,2,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,3,5,4,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,4,2,3,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,4,2,5,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,5,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,4,5,2,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[1,4,5,3,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,2,3,4,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,2,4,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,4,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,4,2,3,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,1,3,4,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,1,3,5,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,1,4,3,6,5] => [3,3]
=> [3]
=> []
=> ? = 1 - 1
[2,1,4,5,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,1,5,3,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,1,5,4,3,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,3,1,4,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,3,1,5,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,3,4,1,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,3,4,5,1,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 1
[2,3,5,1,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,3,5,4,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,4,1,3,6,5] => [3,3]
=> [3]
=> []
=> ? = 2 - 1
[2,4,1,5,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,4,3,1,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,4,3,5,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,4,5,1,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 1
[2,4,5,3,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,5,1,4,3,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,5,3,1,4,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,5,3,4,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,5,4,1,3,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,5,4,3,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,1,5,4,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,1,4,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,1,5,4,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,4,1,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,4,5,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,2,5,1,4,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,2,5,4,1,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,4,2,1,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,4,2,5,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,4,5,2,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,5,1,4,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,5,2,1,4,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,5,2,4,1,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,5,4,1,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,5,4,2,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,1,3,2,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[4,1,3,5,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,1,5,3,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[4,2,1,3,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St001175
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%
Values
[1,2] => [2]
=> []
=> ?
=> ? = 0 - 2
[2,1] => [1,1]
=> [1]
=> []
=> ? = 0 - 2
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 1 - 2
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 1 - 2
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 2 - 2
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 1 - 2
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 1 - 2
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 2 - 2
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 2 - 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 1 - 2
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? = 2 - 2
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? = 1 - 2
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 2
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? = 2 - 2
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[3,1,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 2 - 2
[3,1,4,2,5] => [3,2]
=> [2]
=> []
=> ? = 2 - 2
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[3,4,1,2,5] => [3,2]
=> [2]
=> []
=> ? = 2 - 2
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[4,1,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 2 - 2
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[4,2,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[4,3,1,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[1,2,3,4,6,5] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,3,5,4,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,4,3,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,2,4,5,3,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,5,3,4,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,5,4,3,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,3,2,4,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,3,2,5,4,6] => [4,2]
=> [2]
=> []
=> ? = 1 - 2
[1,3,4,2,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,3,4,5,2,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,3,5,2,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,3,5,4,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,4,2,3,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,4,2,5,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[1,4,3,5,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,4,5,2,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,4,5,3,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,5,2,3,4,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,5,3,2,4,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,5,3,4,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,5,4,2,3,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[2,1,3,4,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,1,3,5,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,1,4,3,6,5] => [3,3]
=> [3]
=> []
=> ? = 1 - 2
[2,1,4,5,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,1,5,3,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,1,5,4,3,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[2,3,1,4,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,1,5,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,4,1,6,5] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,4,5,1,6] => [5,1]
=> [1]
=> []
=> ? = 2 - 2
[2,3,5,1,4,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,5,4,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[2,4,1,3,6,5] => [3,3]
=> [3]
=> []
=> ? = 2 - 2
[2,4,1,5,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,4,3,1,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[2,4,3,5,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[2,4,5,1,3,6] => [4,2]
=> [2]
=> []
=> ? = 2 - 2
[2,4,5,3,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[2,5,1,4,3,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[2,5,3,1,4,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[2,5,3,4,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[2,5,4,1,3,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[2,5,4,3,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[3,1,5,4,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,2,1,4,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,2,1,5,4,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,2,4,1,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,2,4,5,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[3,2,5,1,4,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,2,5,4,1,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,4,2,1,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,4,2,5,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[3,4,5,2,1,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[3,5,1,4,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,5,2,1,4,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,5,2,4,1,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,5,4,1,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[3,5,4,2,1,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 2 - 2
[4,1,3,2,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[4,1,3,5,2,6] => [4,1,1]
=> [1,1]
=> [1]
=> 0 = 2 - 2
[4,1,5,3,2,6] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
[4,2,1,3,6,5] => [3,2,1]
=> [2,1]
=> [1]
=> 0 = 2 - 2
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000264The girth of a graph, which is not a tree. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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