Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001036
(load all 4 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ 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] => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1,0,0]
 => 0
[[1],[2]]
 => [2] => [1,1,0,0]
 => 0
[[1,2,3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1,3],[2]]
 => [2,1] => [1,1,0,0,1,0]
 => 1
[[1,2],[3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1],[2],[3]]
 => [3] => [1,1,1,0,0,0]
 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,3,4],[2]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,2,3],[4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,3],[2,4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2],[4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,2,3,4,5],[6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000340
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1] => [1,0,1,0]
 => 0
[[1],[2]]
 => [2] => [1,1] => [1,0,1,0]
 => 0
[[1,2,3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1],[2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,3,4],[2]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,2,3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,3],[2,4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,3],[2],[4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[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
[[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]
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 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]
 => 1
[[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
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 4
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000691
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ 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
[[1,2]]
 => [2] => [1,1] => 11 => 0
[[1],[2]]
 => [2] => [1,1] => 11 => 0
[[1,2,3]]
 => [3] => [1,1,1] => 111 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => 110 => 1
[[1,2],[3]]
 => [3] => [1,1,1] => 111 => 0
[[1],[2],[3]]
 => [3] => [1,1,1] => 111 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,3,4],[2]]
 => [2,2] => [1,2,1] => 1101 => 2
[[1,2,4],[3]]
 => [3,1] => [1,1,2] => 1110 => 1
[[1,2,3],[4]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,3],[2,4]]
 => [2,2] => [1,2,1] => 1101 => 2
[[1,2],[3,4]]
 => [3,1] => [1,1,2] => 1110 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,1,2] => 1110 => 1
[[1,3],[2],[4]]
 => [2,2] => [1,2,1] => 1101 => 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => 1111 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,3,4,5],[2]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => 11010 => 3
[[1,2,5],[3,4]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,3,4],[2,5]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => 11010 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => 11010 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,2] => 11110 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,2,1] => 11101 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,2,1,1] => 11011 => 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => 11111 => 0
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3,4,5,6],[2]]
 => [2,4] => [1,2,1,1,1] => 110111 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,2,1,1] => 111011 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,2,1] => 111101 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1,1,1,2] => 111110 => 1
[[1,2,3,4,5],[6]]
 => [6] => [1,1,1,1,1,1] => 111111 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,2,2,1] => 110101 => 4
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: St001035
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1,0]
 => ? = 0
[[1,2]]
 => [2] => [1,1] => [1,0,1,0]
 => 0
[[1],[2]]
 => [2] => [1,1] => [1,0,1,0]
 => 0
[[1,2,3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,3],[2]]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1],[2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,2,3,4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,3,4],[2]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,2,3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,3],[2,4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[[1,3],[2],[4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[1,2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,4,5],[2]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[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
[[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]
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 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]
 => 1
[[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
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 4
[[1,2,5,6],[3,4]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000388
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 86%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,3,4,5,6,7,8],[2]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2,4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2,5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2,7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2,8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,7,8],[2,4,6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5,6]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4,7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,7],[2,4,8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5,8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6,8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7,8]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,6,7,8],[2,5],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,6,7,8],[2,4],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,6],[4]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,6],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,4],[6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,7],[4]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,7],[5]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,7],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4],[7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5],[7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,8],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,8],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,8],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,8],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,4],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5],[8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6],[8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,6,7,8],[2],[4],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2],[4],[6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2],[5],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2],[4],[7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2],[5],[7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2],[6],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2],[4],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2],[5],[8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2],[6],[8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2],[7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St000722
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 86%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,3,4,5,6,7,8],[2]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2,4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2,5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2,7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2,8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,7,8],[2,4,6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5,6]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4,7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,7],[2,4,8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5,8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6,8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7,8]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,6,7,8],[2,5],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,6,7,8],[2,4],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,6],[4]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,6],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,4],[6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,7],[4]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,7],[5]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,7],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4],[7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5],[7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,8],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,8],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,8],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,8],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,4],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5],[8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6],[8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,6,7,8],[2],[4],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2],[4],[6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2],[5],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2],[4],[7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2],[5],[7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2],[6],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2],[4],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2],[5],[8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2],[6],[8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2],[7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
Description
The number of different neighbourhoods in a graph.
Matching statistic: St001951
(load all 6 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 86%
Values
[[1]]
 => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,3,4,5,6,7,8],[2]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2,4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2,5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2,7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2,8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,7,8],[2,4,6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5,6]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4,7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,7],[2,4,8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5,8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6,8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7,8]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,6,7,8],[2,5],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,6,7,8],[2,4],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,6],[4]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,6],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,4],[6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,7],[4]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,7],[5]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,7],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4],[7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5],[7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,8],[4]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,8],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,8],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,8],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,4],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5],[8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6],[8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,6,7,8],[2],[4],[5]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2],[4],[6]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2],[5],[6]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2],[4],[7]]
 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2],[5],[7]]
 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2],[6],[7]]
 => [2,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2],[4],[8]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2],[5],[8]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2],[6],[8]]
 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2],[7],[8]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2 + 1
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: St001120
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 86%
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)
 => 1
[[1,2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1],[2],[3]]
 => [3] => [3] => ([],3)
 => 0
[[1,2,3,4]]
 => [4] => [4] => ([],4)
 => 0
[[1,3,4],[2]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,4],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
[[1,2,3],[4]]
 => [4] => [4] => ([],4)
 => 0
[[1,3],[2,4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
[[1,4],[2],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1
[[1,3],[2],[4]]
 => [2,2] => [2,2] => ([(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
[[1,3,4,5],[2]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,2,3,4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,4],[3,5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3],[4,5]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,4,5],[2],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,3,4],[2],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,4],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3],[4],[5]]
 => [5] => [5] => ([],5)
 => 0
[[1,4],[2,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,3],[2,4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1
[[1,4],[2],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3],[2],[4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(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)
 => 2
[[1,2,4,5,6],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1
[[1,2,3,4,5],[6]]
 => [6] => [6] => ([],6)
 => 0
[[1,3,5,6],[2,4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[1,3,4,5,6,7,8],[2]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,5,6,7,8],[2,4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,6,7,8],[2,5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,7,8],[2,6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,6,8],[2,7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,5,6,7],[2,8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,5,6,7,8],[2],[4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,6,7,8],[2],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,7,8],[2],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,6,8],[2],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,5,6,7],[2],[8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,5,7,8],[2,4,6]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[[1,3,4,7,8],[2,5,6]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,6,8],[2,4,7]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,6,8],[2,5,7]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,5,8],[2,6,7]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,6,7],[2,4,8]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,6,7],[2,5,8]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,7],[2,6,8]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,6],[2,7,8]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,6,7,8],[2,5],[4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,6,7,8],[2,4],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,7,8],[2,6],[4]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[[1,3,4,7,8],[2,6],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,7,8],[2,4],[6]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[[1,3,4,7,8],[2,5],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,6,8],[2,7],[4]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,6,8],[2,7],[5]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,5,8],[2,7],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,6,8],[2,4],[7]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,6,8],[2,5],[7]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,5,8],[2,6],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,5,6,7],[2,8],[4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,6,7],[2,8],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,7],[2,8],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,6],[2,8],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,5,6,7],[2,4],[8]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,6,7],[2,5],[8]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,7],[2,6],[8]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,6],[2,7],[8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,6,7,8],[2],[4],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,7,8],[2],[4],[6]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[[1,3,4,7,8],[2],[5],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,5,6,8],[2],[4],[7]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,6,8],[2],[5],[7]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,5,8],[2],[6],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,5,6,7],[2],[4],[8]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,6,7],[2],[5],[8]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,7],[2],[6],[8]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,3,4,5,6],[2],[7],[8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
Description
The length of a longest path in a graph.
Matching statistic: St000071
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 86%
Values
[[1]]
 => [1] => [[1],[]]
 => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 5 = 4 + 1
[[1,3,4,5,6,7,8],[2]]
 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 2 + 1
[[1,3,5,6,7,8],[2,4]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,4,6,7,8],[2,5]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2,7]]
 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 3 + 1
[[1,3,4,5,6,7],[2,8]]
 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 2 + 1
[[1,3,5,6,7,8],[2],[4]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,4,6,7,8],[2],[5]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2],[6]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2],[7]]
 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 3 + 1
[[1,3,4,5,6,7],[2],[8]]
 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 2 + 1
[[1,3,5,7,8],[2,4,6]]
 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5,6]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4,7]]
 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,5,6,7],[2,4,8]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,4,6,7],[2,5,8]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6,8]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7,8]]
 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 3 + 1
[[1,3,6,7,8],[2,5],[4]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,6,7,8],[2,4],[5]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,6],[4]]
 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,6],[5]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,4],[6]]
 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5],[6]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,7],[4]]
 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,7],[5]]
 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,7],[6]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4],[7]]
 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5],[7]]
 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6],[7]]
 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 3 + 1
[[1,3,5,6,7],[2,8],[4]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,4,6,7],[2,8],[5]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,8],[6]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,8],[7]]
 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 3 + 1
[[1,3,5,6,7],[2,4],[8]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,4,6,7],[2,5],[8]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6],[8]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7],[8]]
 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 2 + 1
[[1,3,6,7,8],[2],[4],[5]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2],[4],[6]]
 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2],[5],[6]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2],[4],[7]]
 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2],[5],[7]]
 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2],[6],[7]]
 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 3 + 1
[[1,3,5,6,7],[2],[4],[8]]
 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 4 + 1
[[1,3,4,6,7],[2],[5],[8]]
 => [2,3,3] => [[6,4,2],[3,1]]
 => ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2],[6],[8]]
 => [2,4,2] => [[6,5,2],[4,1]]
 => ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2],[7],[8]]
 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 2 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000453
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 86%
Values
[[1]]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[[1,2]]
 => [2] => [2] => ([],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2] => [2] => ([],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => [4] => [4] => ([],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,2,3,4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3,4]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,5],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2],[3,5],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [6] => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,5,6],[4]]
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 3 = 2 + 1
[[1,2,3,4,6],[5]]
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
 => [6] => [6] => ([],6)
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,3,4,5,6,7,8],[2]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2,4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2,5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2,7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2,8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,6,7,8],[2],[4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7,8],[2],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7,8],[2],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6,8],[2],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,4,5,6,7],[2],[8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,5,7,8],[2,4,6]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5,6]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4,7]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,7],[2,4,8]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5,8]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6,8]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7,8]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,6,7,8],[2,5],[4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,6,7,8],[2,4],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,6],[4]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,6],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2,4],[6]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2,5],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,7],[4]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,7],[5]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,7],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2,4],[7]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5],[7]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,8],[4]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,8],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,8],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,8],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2,4],[8]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2,5],[8]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2,6],[8]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2,7],[8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,3,6,7,8],[2],[4],[5]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,7,8],[2],[4],[6]]
 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,7,8],[2],[5],[6]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,5,6,8],[2],[4],[7]]
 => [2,2,3,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2],[5],[7]]
 => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2],[6],[7]]
 => [2,5,1] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[[1,3,5,6,7],[2],[4],[8]]
 => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,6,7],[2],[5],[8]]
 => [2,3,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,7],[2],[6],[8]]
 => [2,4,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,3,4,5,6],[2],[7],[8]]
 => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
Description
The number of distinct Laplacian eigenvalues of a graph.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.