Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001486
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 1
{{1,3},{2}}
 => [3,2,1] => [2,1] => 3
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => 3
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => 4
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,1] => 3
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,2,1] => 5
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1] => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,2,1] => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1] => 3
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,2,1] => 5
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1] => 3
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,3,1] => 4
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,3,1] => 4
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,1] => 4
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,1,1] => 4
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [2,2,1] => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,1,2,1] => 4
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [4,1] => 3
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,3,1] => 4
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,2,1] => 5
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [5,1] => 3
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,2,1] => 5
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [5,1] => 3
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,2,1] => 5
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [5,1] => 3
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,2,1] => 5
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [5,1] => 3
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,3,1] => 5
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [2,3,1] => 5
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [2,3,1] => 5
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,2,1,1] => 5
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,2,1] => 5
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,2,1] => 5
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [5,1] => 3
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,3,1] => 5
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,1] => 5
{{1,3,4,6},{2,5}}
 => [3,5,4,6,2,1] => [2,3,1] => 5
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [2,3,1] => 5
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [2,3,1] => 5
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,2,1] => 5
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [5,1] => 3
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [3,2,1] => 5
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1] => 3
{{1,3,6},{2},{4,5}}
 => [3,2,6,5,4,1] => [2,1,2,1] => 5
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [3,2,1] => 5
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [2,3,1] => 5
{{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [5,1] => 3
{{1,3},{2},{4,6},{5}}
 => [3,2,1,6,5,4] => [2,1,2,1] => 5
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000777
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,3},{2}}
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,4,6},{2,5}}
 => [3,5,4,6,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
{{1,3,6},{2},{4,5}}
 => [3,2,6,5,4,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,3},{2},{4,6},{5}}
 => [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000453
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,3},{2}}
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,4,6},{2,5}}
 => [3,5,4,6,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
{{1,3,6},{2},{4,5}}
 => [3,2,6,5,4,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
{{1,3},{2},{4,6},{5}}
 => [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
{{1,7},{2,6},{3,4,5}}
 => [7,6,4,5,3,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
{{1,7},{2,6},{3,4},{5}}
 => [7,6,4,3,5,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
{{1},{2,7},{3,4,6},{5}}
 => [1,7,4,6,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
{{1},{2,7},{3,6},{4},{5}}
 => [1,7,6,4,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000455
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
{{1}}
 => [1] => [1] => ([],1)
 => ? = 1 - 3
{{1,3},{2}}
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,4,6},{2,5}}
 => [3,5,4,6,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,3,6},{2},{4,5}}
 => [3,2,6,5,4,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,3},{2},{4,6},{5}}
 => [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,4,6},{2,3,5}}
 => [4,3,5,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1,4,6},{2,3},{5}}
 => [4,3,2,6,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1,6},{2,3},{4,5}}
 => [6,3,2,5,4,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
{{1,6},{2,3},{4},{5}}
 => [6,3,2,4,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1},{2,3,6},{4},{5}}
 => [1,3,6,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
{{1},{2,3},{4,6},{5}}
 => [1,3,2,6,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
{{1,4,6},{2,5},{3}}
 => [4,5,3,6,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,4,6},{2},{3,5}}
 => [4,2,5,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,4,6},{2},{3},{5}}
 => [4,2,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
{{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,4,7},{3,5},{6}}
 => [2,4,5,7,3,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,4},{3,7},{5},{6}}
 => [2,4,7,1,5,6,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,2,7},{3},{4},{5},{6}}
 => [2,7,3,4,5,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,3,5},{2,4,7},{6}}
 => [3,4,5,7,1,6,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,3,7},{2,4},{5},{6}}
 => [3,4,7,2,5,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,3,7},{2,6},{4},{5}}
 => [3,6,7,4,5,2,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,3},{2,7},{4},{5},{6}}
 => [3,7,1,4,5,6,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,4,7},{2,5},{3,6}}
 => [4,5,6,7,2,3,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,4},{2,5},{3,7},{6}}
 => [4,5,7,1,2,6,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
{{1,7},{2},{3},{4},{5},{6}}
 => [7,2,3,4,5,6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
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: St001488
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => [1] => [[1],[]]
 => 1
{{1,3},{2}}
 => [3,2,1] => [2,1] => [[2,2],[1]]
 => 3
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => [[3,3],[2]]
 => 3
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => [[3,3],[2]]
 => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
 => 4
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,1] => [[4,4],[3]]
 => 3
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1] => [[4,4],[3]]
 => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1] => [[4,4],[3]]
 => 3
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1] => [[4,4],[3]]
 => 3
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 4
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [4,1] => [[4,4],[3]]
 => 3
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,2,6},{3,4},{5}}
 => [2,6,4,3,5,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ? = 5
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,3,4,6},{2,5}}
 => [3,5,4,6,2,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
{{1,3,6},{2},{4,5}}
 => [3,2,6,5,4,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [5,1] => [[5,5],[4]]
 => ? = 3
{{1,3},{2},{4,6},{5}}
 => [3,2,1,6,5,4] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
{{1,4,6},{2,3,5}}
 => [4,3,5,6,2,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1,4,6},{2,3},{5}}
 => [4,3,2,6,5,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1,6},{2,3,4},{5}}
 => [6,3,4,2,5,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1},{2,3,4,6},{5}}
 => [1,3,4,6,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ? = 4
{{1,6},{2,3},{4,5}}
 => [6,3,2,5,4,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
{{1},{2,3,6},{4,5}}
 => [1,3,6,5,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
{{1,6},{2,3},{4},{5}}
 => [6,3,2,4,5,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1},{2,3,6},{4},{5}}
 => [1,3,6,4,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1},{2,3},{4,6},{5}}
 => [1,3,2,6,5,4] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
{{1,4,6},{2,5},{3}}
 => [4,5,3,6,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,4,6},{2},{3,5}}
 => [4,2,5,6,3,1] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,4,6},{2},{3},{5}}
 => [4,2,3,6,5,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
{{1,4},{2},{3,6},{5}}
 => [4,2,6,1,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
{{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1},{2,4,6},{3,5}}
 => [1,4,5,6,3,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
{{1,6},{2,4},{3},{5}}
 => [6,4,3,2,5,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.