Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001632
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,2] => ([(0,1)],2)
 => 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[-,+,-] => [2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,+] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,1,-] => [2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
[3,1,2] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,+,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[-,+,-,+] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[-,+,+,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[-,-,+,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 0
[-,+,-,-] => [2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[-,+,4,3] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[-,3,2,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 0
[2,1,+,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,1,-,+] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[2,1,+,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[2,1,-,-] => [2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[2,1,4,3] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[2,3,1,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 0
[3,1,2,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,2,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[3,1,4,2] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[3,+,1,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,+,1,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[3,-,1,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 0
[3,+,4,1] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1,+,2] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1,-,2] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[4,+,1,3] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,+,+,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,+,-,1] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[-,+,+,+,+] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[-,+,-,+,+] => [2,4,5,1,3] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[-,+,+,-,+] => [2,3,5,1,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[-,+,+,+,-] => [2,3,4,1,5] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1
[-,-,+,-,+] => [3,5,1,2,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 0
[-,-,+,+,-] => [3,4,1,2,5] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 1
[-,+,-,-,+] => [2,5,1,3,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
[-,+,-,+,-] => [2,4,1,3,5] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[-,+,+,-,-] => [2,3,1,4,5] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[-,-,-,+,-] => [4,1,2,3,5] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 1
[-,-,+,-,-] => [3,1,2,4,5] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 0
[-,+,-,-,-] => [2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[-,+,+,5,4] => [2,3,5,1,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[-,-,+,5,4] => [3,5,1,2,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 0
[-,+,-,5,4] => [2,5,1,3,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
[-,+,4,3,+] => [2,4,5,1,3] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 2
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 2
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 2
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 2
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,+,1,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,-,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[3,+,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,-,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,+,-,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,+,-,+,+] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,+,-,+] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[-,+,+,+,-] => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[-,-,+,-,+] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,-,+,+,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,-,+,-] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,+,-,-] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,-,-,+,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[-,-,+,-,-] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ? = 0
[-,+,-,-,-] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ? = 2
[-,+,+,5,4] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[-,-,+,5,4] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,+,-,5,4] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,4,3,+] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,4,3,-] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,-,4,3,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[-,+,4,5,3] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,5,3,4] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,5,+,3] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,5,-,3] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,3,2,-,+] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,3,2,+,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,3,2,-,-] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ? = 0
[-,3,2,5,4] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,3,4,2,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[-,4,2,3,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,4,2,5,3] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,4,+,2,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,4,-,2,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,+,1,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,+,+,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[2,1,+,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[3,1,2,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[3,+,1,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1,2,3,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1,+,2,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,+,1,3,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,+,+,1,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,2,3,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,2,+,3,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,+,2,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,+,+,2,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,1,3,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,1,+,3,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,+,1,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,+,+,1,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,3,4,5] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,3,+,4] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,+,3,5] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,+,+,3] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 2
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 2
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 2
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 2
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,+,1,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[3,-,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[3,+,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,-,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,+,-,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,+,-,+,+] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,+,-,+] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[-,+,+,+,-] => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[-,-,+,-,+] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,-,+,+,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,-,+,-] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,+,-,-] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,-,-,+,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[-,-,+,-,-] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ? = 0
[-,+,-,-,-] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ? = 2
[-,+,+,5,4] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1
[-,-,+,5,4] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,+,-,5,4] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,4,3,+] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,4,3,-] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,-,4,3,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[-,+,4,5,3] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,+,5,3,4] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,5,+,3] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[-,+,5,-,3] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
[-,3,2,-,+] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,3,2,+,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,3,2,-,-] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ? = 0
[-,3,2,5,4] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,3,4,2,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[-,4,2,3,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,4,2,5,3] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[-,4,+,2,-] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 1
[-,4,-,2,-] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,+,1,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,+,+,+,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[2,1,+,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[3,1,2,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[3,+,1,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1,2,3,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,1,+,2,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,+,1,3,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,+,+,1,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,2,3,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,2,+,3,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,+,2,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,1,+,+,2,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,1,3,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,1,+,3,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,+,1,4,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[5,+,+,+,1,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,3,4,5] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,3,+,4] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,+,3,5] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[6,1,2,+,+,3] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[-,+] => [2,1] => [1,2] => 1 => 1
[2,1] => [2,1] => [1,2] => 1 => 1
[-,+,+] => [2,3,1] => [1,3,2] => 10 => 1
[-,+,-] => [2,1,3] => [3,1,2] => 00 => ? = 2
[2,1,+] => [2,3,1] => [1,3,2] => 10 => 1
[2,1,-] => [2,1,3] => [3,1,2] => 00 => ? = 2
[3,1,2] => [2,3,1] => [1,3,2] => 10 => 1
[3,+,1] => [2,3,1] => [1,3,2] => 10 => 1
[-,+,+,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[-,+,+,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? = 1
[-,-,+,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? = 0
[-,+,-,-] => [2,1,3,4] => [4,3,1,2] => 000 => ? = 2
[-,+,4,3] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[-,3,2,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? = 0
[2,1,+,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[2,1,-,+] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[2,1,+,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? = 1
[2,1,-,-] => [2,1,3,4] => [4,3,1,2] => 000 => ? = 2
[2,1,4,3] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[2,3,1,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? = 0
[3,1,2,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[3,1,2,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? = 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[3,+,1,+] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[3,+,1,-] => [2,3,1,4] => [4,1,3,2] => 000 => ? = 1
[3,-,1,-] => [3,1,2,4] => [4,2,1,3] => 000 => ? = 0
[3,+,4,1] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,1,+,2] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,1,-,2] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[4,+,1,3] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,+,+,1] => [2,3,4,1] => [1,4,3,2] => 100 => 1
[4,+,-,1] => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2
[-,+,+,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[-,+,-,+,+] => [2,4,5,1,3] => [3,1,5,4,2] => 0000 => ? = 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,1,5,3,2] => 0000 => ? = 1
[-,+,+,+,-] => [2,3,4,1,5] => [5,1,4,3,2] => 0000 => ? = 1
[-,-,+,-,+] => [3,5,1,2,4] => [4,2,1,5,3] => 0000 => ? = 0
[-,-,+,+,-] => [3,4,1,2,5] => [5,2,1,4,3] => 0000 => ? = 1
[-,+,-,-,+] => [2,5,1,3,4] => [4,3,1,5,2] => 0000 => ? = 2
[-,+,-,+,-] => [2,4,1,3,5] => [5,3,1,4,2] => 0000 => ? = 2
[-,+,+,-,-] => [2,3,1,4,5] => [5,4,1,3,2] => 0000 => ? = 1
[-,-,-,+,-] => [4,1,2,3,5] => [5,3,2,1,4] => 0000 => ? = 1
[-,-,+,-,-] => [3,1,2,4,5] => [5,4,2,1,3] => 0000 => ? = 0
[-,+,-,-,-] => [2,1,3,4,5] => [5,4,3,1,2] => 0000 => ? = 2
[-,+,+,5,4] => [2,3,5,1,4] => [4,1,5,3,2] => 0000 => ? = 1
[-,-,+,5,4] => [3,5,1,2,4] => [4,2,1,5,3] => 0000 => ? = 0
[-,+,-,5,4] => [2,5,1,3,4] => [4,3,1,5,2] => 0000 => ? = 2
[-,+,4,3,+] => [2,4,5,1,3] => [3,1,5,4,2] => 0000 => ? = 2
[-,+,4,3,-] => [2,4,1,3,5] => [5,3,1,4,2] => 0000 => ? = 2
[-,-,4,3,-] => [4,1,2,3,5] => [5,3,2,1,4] => 0000 => ? = 1
[-,+,4,5,3] => [2,5,1,3,4] => [4,3,1,5,2] => 0000 => ? = 2
[-,+,5,3,4] => [2,4,5,1,3] => [3,1,5,4,2] => 0000 => ? = 2
[-,+,5,+,3] => [2,4,5,1,3] => [3,1,5,4,2] => 0000 => ? = 2
[-,+,5,-,3] => [2,5,1,3,4] => [4,3,1,5,2] => 0000 => ? = 2
[-,3,2,-,+] => [3,5,1,2,4] => [4,2,1,5,3] => 0000 => ? = 0
[-,3,2,+,-] => [3,4,1,2,5] => [5,2,1,4,3] => 0000 => ? = 1
[-,3,2,-,-] => [3,1,2,4,5] => [5,4,2,1,3] => 0000 => ? = 0
[-,3,2,5,4] => [3,5,1,2,4] => [4,2,1,5,3] => 0000 => ? = 0
[-,3,4,2,-] => [4,1,2,3,5] => [5,3,2,1,4] => 0000 => ? = 1
[-,4,2,3,-] => [3,4,1,2,5] => [5,2,1,4,3] => 0000 => ? = 1
[-,4,2,5,3] => [3,5,1,2,4] => [4,2,1,5,3] => 0000 => ? = 0
[-,4,+,2,-] => [3,4,1,2,5] => [5,2,1,4,3] => 0000 => ? = 1
[-,4,-,2,-] => [4,1,2,3,5] => [5,3,2,1,4] => 0000 => ? = 1
[2,1,+,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[3,1,2,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[3,+,1,+,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[4,1,2,3,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[4,1,+,2,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[4,+,1,3,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[4,+,+,1,+] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,1,2,3,4] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,1,2,+,3] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,1,+,2,4] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,1,+,+,2] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,+,1,3,4] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,+,1,+,3] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,+,+,1,4] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[5,+,+,+,1] => [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.