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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St001395
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(2,3)],4)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(3,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> 0
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
Description
The number of strictly unfriendly partitions of a graph.
A strictly unfriendly partitions of a graph is a two-colouring of its vertices such that every vertex has more neighbours of the other colour than of the same colour.
This statistic returns the number of strictly unfriendly partitions, up to switching the colours. For example, the complete graph on four vertices has three strictly unfriendly partitions: the three set partitions of the vertices into two blocks of size two.
Matching statistic: St000260
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1,1] => [2] => ([],2)
=> ? = 0 + 1
([(1,2)],3)
=> [1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => ([],3)
=> ? = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(2,3)],4)
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(3,4)],5)
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(4,5)],6)
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(3,5),(4,5)],6)
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(2,5),(3,4)],6)
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [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)
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> 1 = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,3,1] => [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)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> 1 = 0 + 1
([(3,6),(4,5)],7)
=> [2,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(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)
=> 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
=> [2,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(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)
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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)
=> 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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)
=> 1 = 0 + 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(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)
=> 1 = 0 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(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)
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> [3,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(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)
=> 1 = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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)
=> 1 = 0 + 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,3] => [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)
=> 1 = 0 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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)
=> 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(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)
=> 1 = 0 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(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)
=> 1 = 0 + 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000259
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1,1] => [2] => ([],2)
=> ? = 0 + 2
([(1,2)],3)
=> [1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 2
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => ([],3)
=> ? = 0 + 2
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
([(2,3)],4)
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
([(1,3),(2,3)],4)
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> ? = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 2
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
([(3,4)],5)
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
([(2,4),(3,4)],5)
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 0 + 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 0 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 2 + 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 0 + 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
([(4,5)],6)
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(3,5),(4,5)],6)
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(2,5),(3,4)],6)
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 1 + 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 + 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 0 + 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 1 + 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? = 0 + 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [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 = 0 + 2
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,3,1] => [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 = 0 + 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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 = 0 + 2
([(3,6),(4,5)],7)
=> [2,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(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 = 0 + 2
([(4,5),(4,6),(5,6)],7)
=> [2,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(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 = 0 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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 = 0 + 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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 = 0 + 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(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 = 0 + 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(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 = 0 + 2
([(1,6),(2,5),(3,4)],7)
=> [3,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(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 = 0 + 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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 = 0 + 2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,1,3] => [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 = 0 + 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => [1,1,2,2,1] => ([(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 = 0 + 2
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(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 = 0 + 2
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(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 = 0 + 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000929
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0
([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0
([(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,2]
=> [2]
=> 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> [3]
=> 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001568
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> ? = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1 = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> 1 = 0 + 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,2]
=> [2]
=> 1 = 0 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> [3]
=> 1 = 0 + 1
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 1 = 0 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1]
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> []
=> ? = 0
([(1,3),(2,3)],4)
=> [2]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> []
=> ? = 0
([(2,4),(3,4)],5)
=> [2]
=> []
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [3]
=> []
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> []
=> ? = 0
([(3,5),(4,5)],6)
=> [2]
=> []
=> ? = 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [3]
=> []
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [1,1]
=> [1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [2,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> [1,1]
=> 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000205
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1]
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> []
=> ? = 0
([(1,3),(2,3)],4)
=> [2]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> []
=> ? = 0
([(2,4),(3,4)],5)
=> [2]
=> []
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [3]
=> []
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> []
=> ? = 0
([(3,5),(4,5)],6)
=> [2]
=> []
=> ? = 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [3]
=> []
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [1,1]
=> [1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [2,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> [1,1]
=> 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
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)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1]
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> []
=> ? = 0
([(1,3),(2,3)],4)
=> [2]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> []
=> ? = 0
([(2,4),(3,4)],5)
=> [2]
=> []
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [3]
=> []
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> []
=> ? = 0
([(3,5),(4,5)],6)
=> [2]
=> []
=> ? = 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [3]
=> []
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [1,1]
=> [1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [2,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> [1,1]
=> 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
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: St000225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1]
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> []
=> ? = 0
([(1,3),(2,3)],4)
=> [2]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> []
=> ? = 0
([(2,4),(3,4)],5)
=> [2]
=> []
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [3]
=> []
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> []
=> ? = 0
([(3,5),(4,5)],6)
=> [2]
=> []
=> ? = 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [3]
=> []
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [1,1]
=> [1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [2,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> [1,1]
=> 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000749
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
([(0,1)],2)
=> [1]
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> [2]
=> []
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> []
=> ? = 0
([(1,3),(2,3)],4)
=> [2]
=> []
=> ? = 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> []
=> ? = 0
([(2,4),(3,4)],5)
=> [2]
=> []
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [3]
=> []
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> []
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> []
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> []
=> ? = 0
([(3,5),(4,5)],6)
=> [2]
=> []
=> ? = 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [3]
=> []
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? = 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> []
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [1,1]
=> [1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [2,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [2]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [2,1,1]
=> [1,1]
=> 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [2]
=> 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [2]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [2]
=> 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1]
=> 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [2]
=> 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [2]
=> 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,1]
=> [1]
=> 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1]
=> [1]
=> 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [3]
=> 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> 0
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000944The 3-degree of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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 $eA$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001490The number of connected components of a skew partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset.
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