Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000422
(load all 4 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => [4] => ([],4)
 => 0
[[1,2,4],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[[1,2,3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,2],[3,4]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => [5] => ([],5)
 => 0
[[1,2,3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,2,3],[4,5]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,2,5],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,2,3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,2],[3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3,4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,4,6],[2,5]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,3,4,5],[2,6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3,4],[5,6]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,2,3,6],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4,5],[2],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3,4],[5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,4],[2,5,6]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,2,3],[4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4],[2,5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,2,6],[3],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3,4],[2],[5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,2,3],[4],[5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[1,2],[3,6],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3],[2,4],[5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000514
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000514: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1] => [1]
 => []
 => ? = 0 - 2
[[1,2]]
 => [2] => [2]
 => []
 => ? = 0 - 2
[[1],[2]]
 => [2] => [2]
 => []
 => ? = 0 - 2
[[1,2,3]]
 => [3] => [3]
 => []
 => ? = 0 - 2
[[1,3],[2]]
 => [2,1] => [2,1]
 => [1]
 => ? = 2 - 2
[[1,2],[3]]
 => [3] => [3]
 => []
 => ? = 0 - 2
[[1],[2],[3]]
 => [3] => [3]
 => []
 => ? = 0 - 2
[[1,2,3,4]]
 => [4] => [4]
 => []
 => ? = 0 - 2
[[1,2,4],[3]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2 - 2
[[1,2,3],[4]]
 => [4] => [4]
 => []
 => ? = 0 - 2
[[1,2],[3,4]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2 - 2
[[1,4],[2],[3]]
 => [3,1] => [3,1]
 => [1]
 => ? = 2 - 2
[[1,2],[3],[4]]
 => [4] => [4]
 => []
 => ? = 0 - 2
[[1],[2],[3],[4]]
 => [4] => [4]
 => []
 => ? = 0 - 2
[[1,2,3,4,5]]
 => [5] => [5]
 => []
 => ? = 0 - 2
[[1,2,3,5],[4]]
 => [4,1] => [4,1]
 => [1]
 => ? = 2 - 2
[[1,2,3,4],[5]]
 => [5] => [5]
 => []
 => ? = 0 - 2
[[1,2,3],[4,5]]
 => [4,1] => [4,1]
 => [1]
 => ? = 2 - 2
[[1,2,5],[3],[4]]
 => [4,1] => [4,1]
 => [1]
 => ? = 2 - 2
[[1,2,3],[4],[5]]
 => [5] => [5]
 => []
 => ? = 0 - 2
[[1,2],[3,5],[4]]
 => [4,1] => [4,1]
 => [1]
 => ? = 2 - 2
[[1,5],[2],[3],[4]]
 => [4,1] => [4,1]
 => [1]
 => ? = 2 - 2
[[1,2],[3],[4],[5]]
 => [5] => [5]
 => []
 => ? = 0 - 2
[[1],[2],[3],[4],[5]]
 => [5] => [5]
 => []
 => ? = 0 - 2
[[1,2,3,4,5,6]]
 => [6] => [6]
 => []
 => ? = 0 - 2
[[1,3,4,5,6],[2]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,2,3,4,6],[5]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,2,3,4,5],[6]]
 => [6] => [6]
 => []
 => ? = 0 - 2
[[1,3,4,6],[2,5]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,3,4,5],[2,6]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,2,3,4],[5,6]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,2,3,6],[4],[5]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,3,4,5],[2],[6]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,2,3,4],[5],[6]]
 => [6] => [6]
 => []
 => ? = 0 - 2
[[1,3,4],[2,5,6]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,2,3],[4,6],[5]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,3,4],[2,5],[6]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,2,6],[3],[4],[5]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,3,4],[2],[5],[6]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,2,3],[4],[5],[6]]
 => [6] => [6]
 => []
 => ? = 0 - 2
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [3,2,1]
 => [2,1]
 => 4 = 6 - 2
[[1,2],[3,6],[4],[5]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,3],[2,4],[5],[6]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [5,1]
 => [1]
 => ? = 2 - 2
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,2],[3],[4],[5],[6]]
 => [6] => [6]
 => []
 => ? = 0 - 2
[[1],[2],[3],[4],[5],[6]]
 => [6] => [6]
 => []
 => ? = 0 - 2
[[1,2,3,4,5,6,7]]
 => [7] => [7]
 => []
 => ? = 0 - 2
[[1,2,4,5,6,7],[3]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,3,4,5,7],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,2,3,4,5,6],[7]]
 => [7] => [7]
 => []
 => ? = 0 - 2
[[1,2,5,6,7],[3,4]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,4,5,6],[3,7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,3,4,5],[6,7]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,4,5,6,7],[2],[3]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,3,4,7],[5],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,2,4,5,6],[3],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,3,4,5],[6],[7]]
 => [7] => [7]
 => []
 => ? = 0 - 2
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,5,6],[3,4,7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,4,5,6],[2,7],[3]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,3,4],[5,7],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,2,5,6],[3,4],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,4,5],[3,6],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,4,5,7],[2],[3],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,4,7],[3],[5],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,3,7],[4],[5],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,4,5,6],[2],[3],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,4,5],[3],[6],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,3,4],[5],[6],[7]]
 => [7] => [7]
 => []
 => ? = 0 - 2
[[1,4,5],[2,6,7],[3]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,5],[3,4,7],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,4],[3,5,7],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,5],[3,4,6],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,4,7],[2,5],[3,6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,4,5],[2,6],[3,7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,5],[3,4],[6,7]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,4],[3,5],[6,7]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,4,7],[2,5],[3],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,7],[3,4],[5],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,4,5],[2,7],[3],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,4],[3,7],[5],[6]]
 => [3,3,1] => [3,3,1]
 => [3,1]
 => 4 = 6 - 2
[[1,2,3],[4,7],[5],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,4,5],[2,6],[3],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,5],[3,4],[6],[7]]
 => [3,4] => [4,3]
 => [3]
 => 2 = 4 - 2
[[1,2,7],[3],[4],[5],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
[[1,2,3],[4],[5],[6],[7]]
 => [7] => [7]
 => []
 => ? = 0 - 2
[[1,2],[3,7],[4],[5],[6]]
 => [6,1] => [6,1]
 => [1]
 => ? = 2 - 2
Description
The number of invariant simple graphs when acting with a permutation of given cycle type.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2
[[1,2],[3]]
 => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => ([],4)
 => 0
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2
[[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,4,6],[2,5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3,4,5],[2,6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4,5],[2],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3,4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1,3,4],[2,5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3,6],[2,4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3,4],[2,6],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2,3],[4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4],[2,5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2,6],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4],[2],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3],[4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1,3],[2,4],[5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2],[3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3],[2,4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,6],[2],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3],[2],[4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => 0
[[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => 0
[[1,2,4,5,6,7],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,2,3,4,5,6],[7]]
 => [7] => ([],7)
 => 0
[[1,2,5,6,7],[3,4]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,4,5,6],[3,7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,3,4,5],[6,7]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,3,4,7],[5],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,2,4,5,6],[3],[7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,3,4,5],[6],[7]]
 => [7] => ([],7)
 => 0
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,5,6],[3,4,7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,4,5,6],[2,7],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,3,4],[5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,2,5,6],[3,4],[7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5],[3,6],[7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,4,5,7],[2],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,4,7],[3],[5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,3,7],[4],[5],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,2,3,4],[5],[6],[7]]
 => [7] => ([],7)
 => 0
[[1,2,3],[4],[5],[6],[7]]
 => [7] => ([],7)
 => 0
[[1,2],[3],[4],[5],[6],[7]]
 => [7] => ([],7)
 => 0
[[1],[2],[3],[4],[5],[6],[7]]
 => [7] => ([],7)
 => 0
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000264
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 28%●distinct values known / distinct values provided: 25%
Values
[[1]]
 => [1] => ([],1)
 => ? = 0 - 3
[[1,2]]
 => [2] => ([],2)
 => ? = 0 - 3
[[1],[2]]
 => [2] => ([],2)
 => ? = 0 - 3
[[1,2,3]]
 => [3] => ([],3)
 => ? = 0 - 3
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 - 3
[[1,2],[3]]
 => [3] => ([],3)
 => ? = 0 - 3
[[1],[2],[3]]
 => [3] => ([],3)
 => ? = 0 - 3
[[1,2,3,4]]
 => [4] => ([],4)
 => ? = 0 - 3
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 3
[[1,2,3],[4]]
 => [4] => ([],4)
 => ? = 0 - 3
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 3
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 3
[[1,2],[3],[4]]
 => [4] => ([],4)
 => ? = 0 - 3
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => ? = 0 - 3
[[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = 0 - 3
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 3
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => ? = 0 - 3
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 3
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 3
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => ? = 0 - 3
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 3
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 3
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => ? = 0 - 3
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => ? = 0 - 3
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = 0 - 3
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => ? = 0 - 3
[[1,3,4,6],[2,5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,3,4,5],[2,6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,3,4,6],[2],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,3,4,5],[2],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,2,3,4],[5],[6]]
 => [6] => ([],6)
 => ? = 0 - 3
[[1,3,4],[2,5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,3,6],[2,4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,3,4],[2,6],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,2,3],[4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,3,4],[2,5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,2,6],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,3,4],[2],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,2,3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 0 - 3
[[1,3],[2,4],[5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 6 - 3
[[1,2],[3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,3],[2,4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,6],[2],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 3
[[1,3],[2],[4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 3
[[1,2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 0 - 3
[[1],[2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 0 - 3
[[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => ? = 0 - 3
[[1,2,4,5,6,7],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4 - 3
[[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2 - 3
[[1,2,3,4,5,6],[7]]
 => [7] => ([],7)
 => ? = 0 - 3
[[1,2,5,6,7],[3,4]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4 - 3
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4,5,6],[3,7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4 - 3
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,5,7],[2],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4,7],[3],[5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,5],[2,6,7],[3]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,5],[3,4,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4],[3,5,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,7],[2,5],[3,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,5],[3,4],[6,7]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4],[3,5],[6,7]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,7],[2,5],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,7],[3,4],[5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,5],[2,7],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2,4],[3,7],[5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4,7],[2],[3],[5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4],[2,5],[3,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,2],[3,4],[5,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
[[1,4],[2,7],[3],[5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 6 - 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001603
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 28%●distinct values known / distinct values provided: 25%
Values
[[1]]
 => [1] => [1] => [1]
 => ? = 0 - 5
[[1,2]]
 => [2] => [1] => [1]
 => ? = 0 - 5
[[1],[2]]
 => [2] => [1] => [1]
 => ? = 0 - 5
[[1,2,3]]
 => [3] => [1] => [1]
 => ? = 0 - 5
[[1,3],[2]]
 => [2,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2],[3]]
 => [3] => [1] => [1]
 => ? = 0 - 5
[[1],[2],[3]]
 => [3] => [1] => [1]
 => ? = 0 - 5
[[1,2,3,4]]
 => [4] => [1] => [1]
 => ? = 0 - 5
[[1,2,4],[3]]
 => [3,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2,3],[4]]
 => [4] => [1] => [1]
 => ? = 0 - 5
[[1,2],[3,4]]
 => [3,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,4],[2],[3]]
 => [3,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2],[3],[4]]
 => [4] => [1] => [1]
 => ? = 0 - 5
[[1],[2],[3],[4]]
 => [4] => [1] => [1]
 => ? = 0 - 5
[[1,2,3,4,5]]
 => [5] => [1] => [1]
 => ? = 0 - 5
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2,3,4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 5
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2,3],[4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 5
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2],[3],[4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 5
[[1],[2],[3],[4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 5
[[1,2,3,4,5,6]]
 => [6] => [1] => [1]
 => ? = 0 - 5
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2,3,4,5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 5
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,3,4,5],[2,6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,3,4],[5,6]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,2,3,6],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,3,4,5],[2],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,3,4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 5
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,2,3],[4,6],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,3,4],[2,5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,2,6],[3],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,3,4],[2],[5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,3],[4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 5
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 1 = 6 - 5
[[1,2],[3,6],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,3],[2,4],[5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2],[3],[4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 5
[[1],[2],[3],[4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 5
[[1,2,3,4,5,6,7]]
 => [7] => [1] => [1]
 => ? = 0 - 5
[[1,2,4,5,6,7],[3]]
 => [3,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,3,4,5,7],[6]]
 => [6,1] => [1,1] => [1,1]
 => ? = 2 - 5
[[1,2,3,4,5,6],[7]]
 => [7] => [1] => [1]
 => ? = 0 - 5
[[1,2,5,6,7],[3,4]]
 => [3,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4,5,6],[3,7]]
 => [3,4] => [1,1] => [1,1]
 => ? = 4 - 5
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,5,7],[2],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4,7],[3],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,5],[2,6,7],[3]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,5],[3,4,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4],[3,5,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,7],[2,5],[3,6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,5],[3,4],[6,7]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4],[3,5],[6,7]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,7],[2,5],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,7],[3,4],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,5],[2,7],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2,4],[3,7],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4,7],[2],[3],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4],[2,5],[3,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,2],[3,4],[5,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
[[1,4],[2,7],[3],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 1 = 6 - 5
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001629
(load all 2 compositions to match this statistic)
Mapping
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 25% ●values known / values provided: 28%●distinct values known / distinct values provided: 25%
Values
[[1]]
 => [[1]]
 => [1] => [1] => ? = 0 - 5
[[1,2]]
 => [[1,2]]
 => [2] => [1] => ? = 0 - 5
[[1],[2]]
 => [[1],[2]]
 => [2] => [1] => ? = 0 - 5
[[1,2,3]]
 => [[1,2,3]]
 => [3] => [1] => ? = 0 - 5
[[1,3],[2]]
 => [[1,2],[3]]
 => [2,1] => [1,1] => ? = 2 - 5
[[1,2],[3]]
 => [[1,3],[2]]
 => [3] => [1] => ? = 0 - 5
[[1],[2],[3]]
 => [[1],[2],[3]]
 => [3] => [1] => ? = 0 - 5
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [4] => [1] => ? = 0 - 5
[[1,2,4],[3]]
 => [[1,2,4],[3]]
 => [2,2] => [2] => ? = 2 - 5
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => [4] => [1] => ? = 0 - 5
[[1,2],[3,4]]
 => [[1,2],[3,4]]
 => [2,2] => [2] => ? = 2 - 5
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [2,2] => [2] => ? = 2 - 5
[[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => [4] => [1] => ? = 0 - 5
[[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => [4] => [1] => ? = 0 - 5
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => [5] => [1] => ? = 0 - 5
[[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => [2,3] => [1,1] => ? = 2 - 5
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => [5] => [1] => ? = 0 - 5
[[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => [2,3] => [1,1] => ? = 2 - 5
[[1,2,5],[3],[4]]
 => [[1,2,5],[3],[4]]
 => [2,3] => [1,1] => ? = 2 - 5
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => [5] => [1] => ? = 0 - 5
[[1,2],[3,5],[4]]
 => [[1,2],[3,5],[4]]
 => [2,3] => [1,1] => ? = 2 - 5
[[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => [2,3] => [1,1] => ? = 2 - 5
[[1,2],[3],[4],[5]]
 => [[1,5],[2],[3],[4]]
 => [5] => [1] => ? = 0 - 5
[[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => [5] => [1] => ? = 0 - 5
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => [6] => [1] => ? = 0 - 5
[[1,3,4,5,6],[2]]
 => [[1,2,3,4,5],[6]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,2,3,4,6],[5]]
 => [[1,2,4,5,6],[3]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => [6] => [1] => ? = 0 - 5
[[1,3,4,6],[2,5]]
 => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,3,4,5],[2,6]]
 => [[1,3,4,5],[2,6]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,2,3,4],[5,6]]
 => [[1,2,5,6],[3,4]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,3,4,6],[2],[5]]
 => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,2,3,6],[4],[5]]
 => [[1,2,5,6],[3],[4]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,3,4,5],[2],[6]]
 => [[1,3,4,5],[2],[6]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,2,3,4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => [6] => [1] => ? = 0 - 5
[[1,3,4],[2,5,6]]
 => [[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,3,6],[2,4],[5]]
 => [[1,2,5],[3,6],[4]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,3,4],[2,6],[5]]
 => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,2,3],[4,6],[5]]
 => [[1,2,6],[3,5],[4]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,3,4],[2,5],[6]]
 => [[1,4,5],[2,6],[3]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,3,6],[2],[4],[5]]
 => [[1,2,5],[3],[4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,2,6],[3],[4],[5]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,3,4],[2],[5],[6]]
 => [[1,4,5],[2],[3],[6]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,2,3],[4],[5],[6]]
 => [[1,5,6],[2],[3],[4]]
 => [6] => [1] => ? = 0 - 5
[[1,3],[2,4],[5,6]]
 => [[1,2],[3,5],[4,6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,3],[2,6],[4],[5]]
 => [[1,2],[3,5],[4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 6 - 5
[[1,2],[3,6],[4],[5]]
 => [[1,2],[3,6],[4],[5]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,3],[2,4],[5],[6]]
 => [[1,5],[2,6],[3],[4]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => [2,4] => [1,1] => ? = 2 - 5
[[1,3],[2],[4],[5],[6]]
 => [[1,5],[2],[3],[4],[6]]
 => [5,1] => [1,1] => ? = 4 - 5
[[1,2],[3],[4],[5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [6] => [1] => ? = 0 - 5
[[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [6] => [1] => ? = 0 - 5
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => [7] => [1] => ? = 0 - 5
[[1,2,4,5,6,7],[3]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1] => ? = 4 - 5
[[1,2,3,4,5,7],[6]]
 => [[1,2,4,5,6,7],[3]]
 => [2,5] => [1,1] => ? = 2 - 5
[[1,2,3,4,5,6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => [7] => [1] => ? = 0 - 5
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => [1,1] => ? = 4 - 5
[[1,2,4,5,7],[3,6]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4,5,6],[3,7]]
 => [[1,3,4,5,7],[2,6]]
 => [5,2] => [1,1] => ? = 4 - 5
[[1,2,4,5,7],[3],[6]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,5,7],[3,4,6]]
 => [[1,2,4,5],[3,6,7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4,5],[3,6,7]]
 => [[1,2,5,7],[3,4,6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,5,7],[2,6],[3]]
 => [[1,2,4,5],[3,6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,5,7],[3,4],[6]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4,7],[3,5],[6]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4,5],[3,7],[6]]
 => [[1,2,5,7],[3,4],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,5,7],[2],[3],[6]]
 => [[1,2,4,5],[3],[6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4,7],[3],[5],[6]]
 => [[1,2,5,7],[3],[4],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,5],[2,6,7],[3]]
 => [[1,2,5],[3,4,6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,5],[3,4,7],[6]]
 => [[1,2,5],[3,4,7],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4],[3,5,7],[6]]
 => [[1,2,5],[3,6,7],[4]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,7],[2,5],[3,6]]
 => [[1,2,5],[3,6],[4,7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,5],[3,4],[6,7]]
 => [[1,2,5],[3,4],[6,7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4],[3,5],[6,7]]
 => [[1,2,7],[3,5],[4,6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,7],[2,5],[3],[6]]
 => [[1,2,5],[3,6],[4],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,7],[3,4],[5],[6]]
 => [[1,2,5],[3,7],[4],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,5],[2,7],[3],[6]]
 => [[1,2,5],[3,4],[6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2,4],[3,7],[5],[6]]
 => [[1,2,7],[3,5],[4],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4,7],[2],[3],[5],[6]]
 => [[1,2,5],[3],[4],[6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4],[2,5],[3,7],[6]]
 => [[1,2],[3,5],[4,6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,2],[3,4],[5,7],[6]]
 => [[1,2],[3,5],[4,7],[6]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
[[1,4],[2,7],[3],[5],[6]]
 => [[1,2],[3,5],[4],[6],[7]]
 => [2,3,2] => [1,1,1] => 1 = 6 - 5
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001604
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 28%●distinct values known / distinct values provided: 25%
Values
[[1]]
 => [1] => [1] => [1]
 => ? = 0 - 6
[[1,2]]
 => [2] => [1] => [1]
 => ? = 0 - 6
[[1],[2]]
 => [2] => [1] => [1]
 => ? = 0 - 6
[[1,2,3]]
 => [3] => [1] => [1]
 => ? = 0 - 6
[[1,3],[2]]
 => [2,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2],[3]]
 => [3] => [1] => [1]
 => ? = 0 - 6
[[1],[2],[3]]
 => [3] => [1] => [1]
 => ? = 0 - 6
[[1,2,3,4]]
 => [4] => [1] => [1]
 => ? = 0 - 6
[[1,2,4],[3]]
 => [3,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2,3],[4]]
 => [4] => [1] => [1]
 => ? = 0 - 6
[[1,2],[3,4]]
 => [3,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,4],[2],[3]]
 => [3,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2],[3],[4]]
 => [4] => [1] => [1]
 => ? = 0 - 6
[[1],[2],[3],[4]]
 => [4] => [1] => [1]
 => ? = 0 - 6
[[1,2,3,4,5]]
 => [5] => [1] => [1]
 => ? = 0 - 6
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2,3,4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 6
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2,3],[4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 6
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2],[3],[4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 6
[[1],[2],[3],[4],[5]]
 => [5] => [1] => [1]
 => ? = 0 - 6
[[1,2,3,4,5,6]]
 => [6] => [1] => [1]
 => ? = 0 - 6
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2,3,4,5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 6
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,3,4,5],[2,6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,3,4],[5,6]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,2,3,6],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,3,4,5],[2],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,3,4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 6
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,2,3],[4,6],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,3,4],[2,5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,2,6],[3],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,3,4],[2],[5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,3],[4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 6
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => [1,1,1]
 => 0 = 6 - 6
[[1,2],[3,6],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,3],[2,4],[5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2],[3],[4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 6
[[1],[2],[3],[4],[5],[6]]
 => [6] => [1] => [1]
 => ? = 0 - 6
[[1,2,3,4,5,6,7]]
 => [7] => [1] => [1]
 => ? = 0 - 6
[[1,2,4,5,6,7],[3]]
 => [3,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,3,4,5,7],[6]]
 => [6,1] => [1,1] => [1,1]
 => ? = 2 - 6
[[1,2,3,4,5,6],[7]]
 => [7] => [1] => [1]
 => ? = 0 - 6
[[1,2,5,6,7],[3,4]]
 => [3,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4,5,6],[3,7]]
 => [3,4] => [1,1] => [1,1]
 => ? = 4 - 6
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,5,7],[2],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4,7],[3],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,5],[2,6,7],[3]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,5],[3,4,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4],[3,5,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,7],[2,5],[3,6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,5],[3,4],[6,7]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4],[3,5],[6,7]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,7],[2,5],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,7],[3,4],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,5],[2,7],[3],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2,4],[3,7],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4,7],[2],[3],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4],[2,5],[3,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,2],[3,4],[5,7],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
[[1,4],[2,7],[3],[5],[6]]
 => [3,3,1] => [2,1] => [2,1]
 => 0 = 6 - 6
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001703
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001703: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1] => ([],1)
 => 0
[[1,2]]
 => [2] => ([],2)
 => 0
[[1],[2]]
 => [2] => ([],2)
 => 0
[[1,2,3]]
 => [3] => ([],3)
 => 0
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1,2],[3]]
 => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => ([],4)
 => 0
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 0
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = 0
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => ? = 0
[[1,3,4,6],[2,5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3,4,5],[2,6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4,5],[2],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3,4],[5],[6]]
 => [6] => ([],6)
 => ? = 0
[[1,3,4],[2,5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3,6],[2,4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3,4],[2,6],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2,3],[4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4],[2,5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2,6],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3,4],[2],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2,3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 0
[[1,3],[2,4],[5,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
[[1,2],[3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3],[2,4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,6],[2],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[[1,3],[2],[4],[5],[6]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
[[1,2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 0
[[1],[2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 0
[[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => ? = 0
[[1,2,4,5,6,7],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,2,3,4,5,6],[7]]
 => [7] => ([],7)
 => ? = 0
[[1,2,5,6,7],[3,4]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,4,5,6],[3,7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,3,4,5],[6,7]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,3,4,7],[5],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
[[1,2,4,5,6],[3],[7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,3,4,5],[6],[7]]
 => [7] => ([],7)
 => ? = 0
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,5,6],[3,4,7]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,4,5,6],[2,7],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 4
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[[1,2,3,4],[5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2
Description
The villainy of a graph. The villainy of a permutation of a proper coloring $c$ of a graph is the minimal Hamming distance between $c$ and a proper coloring. The villainy of a graph is the maximal villainy of a permutation of a proper coloring.
Matching statistic: St000950
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000950: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => [1] => [1] => [1,0]
 => 2 = 0 + 2
[[1,2]]
 => [2] => [1,1] => [1,0,1,0]
 => 2 = 0 + 2
[[1],[2]]
 => [2] => [1,1] => [1,0,1,0]
 => 2 = 0 + 2
[[1,2,3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,3],[2]]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
[[1],[2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2,3,4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2,4],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2,3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2],[3,4]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,4],[2],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 2 + 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,3,4,5,6],[2]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,2,3,4,5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,3,4,5],[2,6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3,4],[5,6]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,3,6],[4],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,3,4,5],[2],[6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3,4],[5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,3],[4,6],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,3,4],[2,5],[6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,6],[3],[4],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,3,4],[2],[5],[6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3],[4],[5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2],[3,6],[4],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,3],[2,4],[5],[6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2],[3],[4],[5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1],[2],[3],[4],[5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,2,3,4,5,6,7]]
 => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,2,4,5,6,7],[3]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3,4,5,7],[6]]
 => [6,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,2,3,4,5,6],[7]]
 => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,2,5,6,7],[3,4]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,4,5,6],[3,7]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3,4,5],[6,7]]
 => [6,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,4,5,6,7],[2],[3]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,3,4,7],[5],[6]]
 => [6,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
[[1,2,4,5,6],[3],[7]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,3,4,5],[6],[7]]
 => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,5,6],[3,4,7]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,4,5,6],[2,7],[3]]
 => [3,4] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 2
[[1,2,3,4],[5,7],[6]]
 => [6,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 2
Description
Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.
Matching statistic: St000699
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000699: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 25%
Values
[[1]]
 => [1] => [1] => ([],1)
 => ? = 0 - 5
[[1,2]]
 => [2] => [1] => ([],1)
 => ? = 0 - 5
[[1],[2]]
 => [2] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,3]]
 => [3] => [1] => ([],1)
 => ? = 0 - 5
[[1,3],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2],[3]]
 => [3] => [1] => ([],1)
 => ? = 0 - 5
[[1],[2],[3]]
 => [3] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,3,4]]
 => [4] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,4],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2,3],[4]]
 => [4] => [1] => ([],1)
 => ? = 0 - 5
[[1,2],[3,4]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,4],[2],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? = 0 - 5
[[1],[2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,3,4,5]]
 => [5] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2,3,4],[5]]
 => [5] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2,3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 0 - 5
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 0 - 5
[[1],[2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 0 - 5
[[1,2,3,4,5,6]]
 => [6] => [1] => ([],1)
 => ? = 0 - 5
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,2,3,4,5],[6]]
 => [6] => [1] => ([],1)
 => ? = 0 - 5
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,3,4,5],[2,6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,2,3,4],[5,6]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,2,3,6],[4],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,3,4,5],[2],[6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,2,3,4],[5],[6]]
 => [6] => [1] => ([],1)
 => ? = 0 - 5
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,2,3],[4,6],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,3,4],[2,5],[6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,2,6],[3],[4],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,3,4],[2],[5],[6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,2,3],[4],[5],[6]]
 => [6] => [1] => ([],1)
 => ? = 0 - 5
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ? = 6 - 5
[[1,2],[3,6],[4],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,3],[2,4],[5],[6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,6],[2],[3],[4],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 5
[[1,3],[2],[4],[5],[6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 4 - 5
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,5,7],[2,6],[3]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,5,7],[3,4],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4,7],[3,5],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4,5],[3,7],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,5,7],[2],[3],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4,7],[3],[5],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,5],[2,6,7],[3]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,5],[3,4,7],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4],[3,5,7],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,7],[2,5],[3,6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,5],[3,4],[6,7]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4],[3,5],[6,7]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,7],[2,5],[3],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,7],[3,4],[5],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,5],[2,7],[3],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2,4],[3,7],[5],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4,7],[2],[3],[5],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4],[2,5],[3,7],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,2],[3,4],[5,7],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
[[1,4],[2,7],[3],[5],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 6 - 5
Description
The toughness times the least common multiple of 1,...,n-1 of a non-complete graph. A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$. The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero. This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.
The following 1 statistic also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.