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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
Mp00317: Integer partitions —odd parts⟶ 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]
=> 1 => [1] => ([],1)
=> 0
[2]
=> 0 => [1] => ([],1)
=> 0
[3]
=> 1 => [1] => ([],1)
=> 0
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4]
=> 0 => [1] => ([],1)
=> 0
[5]
=> 1 => [1] => ([],1)
=> 0
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6]
=> 0 => [1] => ([],1)
=> 0
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[7]
=> 1 => [1] => ([],1)
=> 0
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[8]
=> 0 => [1] => ([],1)
=> 0
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[9]
=> 1 => [1] => ([],1)
=> 0
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[10]
=> 0 => [1] => ([],1)
=> 0
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[11]
=> 1 => [1] => ([],1)
=> 0
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001093
Mp00317: Integer partitions —odd parts⟶ 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]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[2]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[3]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[5]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[6]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[7]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[8]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[9]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[10]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[11]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 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: St000455
Mp00317: Integer partitions —odd parts⟶ 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: 68%●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: 68%●distinct values known / distinct values provided: 67%
Values
[1]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[2]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[3]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[4]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[5]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[6]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[7]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[8]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[9]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[10]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[11]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,4,3]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,4,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,3,3,2]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,2,2,2,2,1]
=> 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[12]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[9,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[7,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[7,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[7,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[6,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[5,5,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[5,4,3]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[5,4,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[5,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,3,3,2]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[4,3,2,2,1]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,3,3,2,1]
=> 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,2,2,2,2,1]
=> 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[13]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[12,1]
=> 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[11,2]
=> 10 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[5,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,3,3,2,1]
=> 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,3,2,2,2,1]
=> 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[14]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[9,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[7,4,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[7,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[6,3,3,2]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[6,3,2,2,1]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,4,4,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[5,4,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,2,2,2,2,1]
=> 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[4,3,2,2,2,1]
=> 010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,3,3,2,2,1]
=> 111001 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,2,2,2,2,2,1]
=> 1000001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[15]
=> 1 => [1] => ([],1)
=> ? = 0 - 2
[7,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[6,3,3,2,1]
=> 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,5,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[5,3,2,2,2,1]
=> 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[4,3,3,3,2]
=> 01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,3,3,2,2,1]
=> 011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,3,2,2,2,2,1]
=> 1100001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[16]
=> 0 => [1] => ([],1)
=> ? = 0 - 2
[11,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[9,4,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 2
[9,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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: St001199
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> ? = 0 - 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ?
=> ? = 2 - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 - 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,2,1,4,5,6,7,8,9] => ?
=> ? = 1 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,1,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,2,1,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,7,3,1,2,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[5,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,1,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,1,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[5,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,1,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,1,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[5,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,1,2,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[5,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,1,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[5,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[5,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[5,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> ? = 0 - 2
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 0 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 - 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 2
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 2
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 - 2
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ?
=> ? = 2 - 2
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 2
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 2
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 2
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 - 2
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,2,1,4,5,6,7,8,9] => ?
=> ? = 1 - 2
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,1,3,4,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 2
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 2
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,2,1,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 2
[5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,7,3,1,2,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 2
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 2
[4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[5,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,1,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,1,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[5,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[5,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,1,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[5,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,1,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 0 = 2 - 2
[3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[5,5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0 = 2 - 2
[5,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,1,2,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[5,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,1,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[5,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[5,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[5,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 0 = 2 - 2
[4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001637
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001637: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Mp00262: Binary words —poset of factors⟶ Posets
St001637: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Values
[1]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[3]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[5]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[6]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[7]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[8]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[9]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[8,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 + 1
[10]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 + 1
[11]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[10,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[4,4,3]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[4,4,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[4,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 + 1
[3,3,3,2]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
[2,2,2,2,2,1]
=> 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 + 1
[12]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[7,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[5,5,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 + 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 + 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 2 + 1
[13]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[12,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[10,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[10,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[8,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[6,6,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[14]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[15]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[14,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[12,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[10,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,7]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[16]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[17]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[16,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[15,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[14,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[13,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[12,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001668: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Mp00262: Binary words —poset of factors⟶ Posets
St001668: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Values
[1]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[3]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[5]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[6]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[7]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[8]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[9]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[8,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 + 1
[10]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 + 1
[11]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[10,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[4,4,3]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[4,4,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[4,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 + 1
[3,3,3,2]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
[2,2,2,2,2,1]
=> 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 + 1
[12]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[7,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[5,5,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 + 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 + 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 2 + 1
[13]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[12,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[10,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[10,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[8,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[6,6,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[14]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[15]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[14,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[12,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[10,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,7]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[16]
=> 0 => ([(0,1)],2)
=> 1 = 0 + 1
[17]
=> 1 => ([(0,1)],2)
=> 1 = 0 + 1
[16,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[15,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[14,3]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[13,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[12,5]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001232
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[1]
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> ? = 0
[3]
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> ? = 0
[5]
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[6]
=> []
=> []
=> ? = 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[7]
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[8]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[9]
=> []
=> []
=> ? = 0
[8,1]
=> [1]
=> [1,0,1,0]
=> 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[10]
=> []
=> []
=> ? = 0
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[11]
=> []
=> []
=> ? = 0
[10,1]
=> [1]
=> [1,0,1,0]
=> 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[8,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[7,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[6,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[4,4,3]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[4,4,2,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[4,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[3,3,3,2]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[12]
=> []
=> []
=> ? = 0
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? = 1
[7,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[7,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[7,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[5,5,2]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[5,4,3]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[5,4,2,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[12,1]
=> [1]
=> [1,0,1,0]
=> 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[10,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[9,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[8,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[14,1]
=> [1]
=> [1,0,1,0]
=> 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[12,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[11,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[10,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[9,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[16,1]
=> [1]
=> [1,0,1,0]
=> 1
[15,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[14,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[13,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[12,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[11,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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