Your data matches 52 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001486
(load all 5 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley 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
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1] => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1] => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1] => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [4,1] => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [4,1] => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [4,1] => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1] => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [4,1] => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [4,1] => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [4,1] => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [3,3,1] => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [4,2,1] => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [6,1] => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [3,3,1] => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [4,2,1] => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,1] => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [3,3,1] => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [4,2,1] => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [4,2,1] => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [3,3,1] => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [4,2,1] => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [3,3,1] => 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: St000388
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St000452
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000452: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
Description
The number of distinct eigenvalues of a graph.
Matching statistic: St000453
(load all 3 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000722
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000777
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley 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
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [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]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[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)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[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)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[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)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[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)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[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)
 => 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001352
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001951
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => ([],1)
 => 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
Description
The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$. In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.
Matching statistic: St000691
(load all 6 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => 1 => 0 = 1 - 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1] => 1001 => 2 = 3 - 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1] => 1001 => 2 = 3 - 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1] => 1001 => 2 = 3 - 1
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [4,1] => 10001 => 2 = 3 - 1
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [3,2,1] => 100101 => 4 = 5 - 1
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [5,1] => 100001 => 2 = 3 - 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 1000001 => 2 = 3 - 1
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [3,3,1] => 1001001 => 4 = 5 - 1
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [4,2,1] => 1000101 => 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [6,1] => 1000001 => 2 = 3 - 1
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 1000001 => 2 = 3 - 1
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [3,3,1] => 1001001 => 4 = 5 - 1
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [4,2,1] => 1000101 => 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,1] => 1000001 => 2 = 3 - 1
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 1000001 => 2 = 3 - 1
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [3,3,1] => 1001001 => 4 = 5 - 1
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [4,2,1] => 1000101 => 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1] => 1000001 => 2 = 3 - 1
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [4,2,1] => 1000101 => 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [3,3,1] => 1001001 => 4 = 5 - 1
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [4,2,1] => 1000101 => 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [3,3,1] => 1001001 => 4 = 5 - 1
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001036
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [1] => [1,0]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,4],[3,5]]
 => [[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,4],[3],[5]]
 => [[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2],[3,5],[4]]
 => [[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3,5,6],[4]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,5,6],[2,4]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3,5],[4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,5,6],[2],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,5,6],[3],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3,5],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2,6],[4]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,5],[3,6],[4]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2,4],[6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,5],[3],[4],[6]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3],[2,4],[5,6]]
 => [[1,2,4,6],[3,5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2],[3,4],[5,6]]
 => [[1,2,3,4],[5,6]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,6],[3],[5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2],[3,6],[4],[5]]
 => [[1,2,3,6],[4],[5]]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,4,6,7],[2,5]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,4,6,7],[3,5]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,4,6,7],[2],[5]]
 => [[1,2,4,5,7],[3],[6]]
 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,4,6,7],[3],[5]]
 => [[1,2,3,5,7],[4],[6]]
 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,4,6],[2,5,7]]
 => [[1,2,4,5,7],[3,6]]
 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,4,6],[3,5,7]]
 => [[1,2,3,5,7],[4,6]]
 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,4,6,7],[2,5],[3]]
 => [[1,2,5,7],[3,6],[4]]
 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
 => [[1,2,4,5],[3,7],[6]]
 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,2,6,7],[3,5],[4]]
 => [[1,2,3,5],[4,7],[6]]
 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
 => [[1,2,4,7],[3,5],[6]]
 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => 4 = 5 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001120The length of a longest path in a graph. St001458The rank of the adjacency matrix of a graph. St000402Half the size of the symmetry class of a permutation. St000983The length of the longest alternating subword. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000264The girth of a graph, which is not a tree. St001644The dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001760The number of prefix or suffix reversals needed to sort a permutation. St000638The number of up-down runs of a permutation. St001388The number of non-attacking neighbors of a permutation. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St000647The number of big descents of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St000837The number of ascents of distance 2 of a permutation. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001703The villainy of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001569The maximal modular displacement of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001948The number of augmented double ascents of a permutation. St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St001545The second Elser number of a connected graph. St001557The number of inversions of the second entry of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation.