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Your data matches 181 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1]
=> [1,0,1,0]
=> 1
[[1,2]]
=> [2] => [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[2]]
=> [2] => [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2,3]]
=> [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3,4]]
=> [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,4],[2]]
=> [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1],[2],[3],[4]]
=> [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,3,4,5]]
=> [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,3,4],[5]]
=> [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,3],[4],[5]]
=> [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,3,4,5,6]]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[[1,2,4,5,6],[3]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4,5],[6]]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,4,5],[3,6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4,5],[3],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,5],[3,4,6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,4,5],[2,6],[3]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,5],[3,4],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,4],[3,5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3],[5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,4],[2,5],[3],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2],[3,4],[5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000329
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [1] => [1]
=> [1,0]
=> 0 = 1 - 1
[[1,2]]
=> [2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[3]]
=> [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4],[5]]
=> [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,4,5,6],[3]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,5,6],[3,4]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5],[3,6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,4,5,6],[2],[3]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,4,5],[3],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3,4],[5],[6]]
=> [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4,6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,4,5],[2,6],[3]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[3,5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,4,5],[2],[3],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3],[4],[5],[6]]
=> [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,2],[3,4],[5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5],[6]]
=> [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 7 - 1
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => [6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => [6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => [6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => [6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,4,9],[5,6],[7,8]]
=> [5,2,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,2,3,4,5,10],[6,7],[8,9]]
=> [6,2,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,10],[4,5],[6,7],[8,9]]
=> [4,2,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,5,6,7,8,9],[2],[4]]
=> [2,2,5] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,6,8,9],[2],[3],[4],[5],[7]]
=> [5,2,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,6],[2,8],[3],[4],[5],[7],[9]]
=> [5,2,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,5,6,7,8,9,10],[2],[4]]
=> [2,2,6] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,2,5,7,9,10],[3,6,8],[4]]
=> [4,2,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,7,9,10],[2],[3],[4],[5],[6],[8]]
=> [6,2,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,5],[2,7],[3,9],[4],[6],[8],[10]]
=> [4,2,2,2] => [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[1,7],[2,9],[3],[4],[5],[6],[8],[10]]
=> [6,2,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,4,5,6],[2,7,8,9]]
=> [2,5,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,4,5,6],[2,8,9],[7]]
=> [2,5,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,4,5,6],[2,8],[7,9]]
=> [2,5,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,4,5,6],[2,8],[7],[9]]
=> [2,5,2] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3],[2,5],[4],[6],[7],[8],[9]]
=> [2,2,5] => [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,4,5,6,7],[2,8,9,10]]
=> [2,6,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,4,5,6,7],[2,9,10],[8]]
=> [2,6,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,3,4,5,6,7],[2,9],[8,10]]
=> [2,6,2] => [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 83%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [1] => [1,0]
=> 1
[[1,2]]
=> [2] => [1,1,0,0]
=> 1
[[1],[2]]
=> [2] => [1,1,0,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 1
[[1,2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 1
[[1],[2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[[1,3,4],[2]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[[1,3],[2,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[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]
=> 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,2,4,5],[3,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,4,5],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3,4,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,4,5],[2,6],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3,4],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,2,4],[3,5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,4],[3],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,4],[2,5],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2],[3,4],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 5
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 7
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => ?
=> ? = 7
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => ?
=> ? = 5
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => ?
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => ?
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => ?
=> ? = 7
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[1,4,7],[2,5,8],[3,6,9]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,2,3,4,9],[5,6],[7,8]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,2,5],[3,4,8],[6,7,9]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,2,3,4,5,10],[6,7],[8,9]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[[1,2,3,10],[4,5],[6,7],[8,9]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[1,3,5,6,7,8,9],[2],[4]]
=> [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[1,2,4,7,8,9],[3,5],[6]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,6,8,9],[2],[3],[4],[5],[7]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000288
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 83%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [[1]]
=> [1] => 1 => 1
[[1,2]]
=> [[1,2]]
=> [2] => 10 => 1
[[1],[2]]
=> [[1],[2]]
=> [2] => 10 => 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => 100 => 1
[[1,2],[3]]
=> [[1,3],[2]]
=> [3] => 100 => 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [3] => 100 => 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => 1000 => 1
[[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [4] => 1000 => 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [4] => 1000 => 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [4] => 1000 => 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => 10000 => 1
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [5] => 10000 => 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [5] => 10000 => 1
[[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> [5] => 10000 => 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [5] => 10000 => 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => 100000 => 1
[[1,2,4,5,6],[3]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => 100010 => 2
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [6] => 100000 => 1
[[1,3,5,6],[2,4]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => 100101 => 3
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 100010 => 2
[[1,2,4,5],[3,6]]
=> [[1,3,4,6],[2,5]]
=> [4,2] => 100010 => 2
[[1,4,5,6],[2],[3]]
=> [[1,2,3,4],[5],[6]]
=> [4,2] => 100010 => 2
[[1,3,5,6],[2],[4]]
=> [[1,2,3,5],[4],[6]]
=> [3,2,1] => 100101 => 3
[[1,2,4,5],[3],[6]]
=> [[1,3,4,6],[2],[5]]
=> [4,2] => 100010 => 2
[[1,2,3,4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [6] => 100000 => 1
[[1,3,5],[2,4,6]]
=> [[1,3,5],[2,4,6]]
=> [3,2,1] => 100101 => 3
[[1,2,5],[3,4,6]]
=> [[1,3,4],[2,5,6]]
=> [4,2] => 100010 => 2
[[1,3,6],[2,5],[4]]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 100101 => 3
[[1,4,5],[2,6],[3]]
=> [[1,3,4],[2,5],[6]]
=> [4,2] => 100010 => 2
[[1,3,5],[2,6],[4]]
=> [[1,3,5],[2,4],[6]]
=> [3,2,1] => 100101 => 3
[[1,3,5],[2,4],[6]]
=> [[1,3,5],[2,6],[4]]
=> [3,2,1] => 100101 => 3
[[1,2,5],[3,4],[6]]
=> [[1,3,4],[2,6],[5]]
=> [4,2] => 100010 => 2
[[1,2,4],[3,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [4,2] => 100010 => 2
[[1,4,5],[2],[3],[6]]
=> [[1,3,4],[2],[5],[6]]
=> [4,2] => 100010 => 2
[[1,3,5],[2],[4],[6]]
=> [[1,3,5],[2],[4],[6]]
=> [3,2,1] => 100101 => 3
[[1,2,4],[3],[5],[6]]
=> [[1,4,6],[2],[3],[5]]
=> [4,2] => 100010 => 2
[[1,2,3],[4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [6] => 100000 => 1
[[1,4],[2,5],[3,6]]
=> [[1,4],[2,5],[3,6]]
=> [4,2] => 100010 => 2
[[1,3],[2,5],[4,6]]
=> [[1,3],[2,5],[4,6]]
=> [3,2,1] => 100101 => 3
[[1,4],[2,5],[3],[6]]
=> [[1,4],[2,5],[3],[6]]
=> [4,2] => 100010 => 2
[[1,3],[2,5],[4],[6]]
=> [[1,3],[2,5],[4],[6]]
=> [3,2,1] => 100101 => 3
[[1,2],[3,4],[5],[6]]
=> [[1,4],[2,6],[3],[5]]
=> [4,2] => 100010 => 2
[[1,4],[2],[3],[5],[6]]
=> [[1,4],[2],[3],[5],[6]]
=> [4,2] => 100010 => 2
[[1,2],[3],[4],[5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [6] => 100000 => 1
[[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6] => 100000 => 1
[[1,2,3,5,6,7,8],[4]]
=> [[1,2,3,4,5,7,8],[6]]
=> [5,3] => 10000100 => 2
[[1,3,5,6,7,8],[2,4]]
=> [[1,2,3,4,5,7],[6,8]]
=> [5,2,1] => 10000101 => 3
[[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [3,4,2,1] => 1001000101 => ? = 5
[[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [3,2,4,1] => 1001010001 => ? = 5
[[1,2,3,5,7],[4,6,8,9,10]]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [3,2,2,3] => 1001010100 => ? = 5
[[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [6,4] => 1000001000 => ? = 2
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [3,4,2,2,1] => 100100010101 => ? = 7
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [5,4,2,1] => 100001000101 => ? = 4
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [3,2,4,2,1] => 100101000101 => ? = 7
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [3,4,4,1] => 100100010001 => ? = 4
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [3,4,4,1] => 100100010001 => ? = 4
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [5,2,4,1] => 100001010001 => ? = 4
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [3,2,2,4,1] => 100101010001 => ? = 7
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [[1,3,4,7,9,10],[2,5,6,8,11,12]]
=> [4,3,3,2] => 100010010010 => ? = 4
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [[1,2,3,4,9,10],[5,6,7,8,11,12]]
=> [4,6,2] => 100010000010 => ? = 5
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [[1,3,4,6,7,10],[2,5,8,9,11,12]]
=> [4,3,3,2] => 100010010010 => ? = 4
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [[1,2,3,4,7,10],[5,6,8,9,11,12]]
=> [4,3,3,2] => 100010010010 => ? = 4
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [3,2,4,3] => 100101000100 => ? = 4
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [5,4,3] => 100001000100 => ? = 3
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [3,2,4,3] => 100101000100 => ? = 4
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [3,4,2,3] => 100100010100 => ? = 4
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [3,4,2,3] => 100100010100 => ? = 4
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [5,2,2,3] => 100001010100 => ? = 4
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [3,2,2,2,3] => 100101010100 => ? = 7
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [[1,2,3,4,5,9],[6,7,8,10,11,12]]
=> [5,4,3] => 100001000100 => ? = 3
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [[1,2,3,4,6,7],[5,8,9,10,11,12]]
=> [4,3,5] => 100010010000 => ? = 5
[[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [3,2,4] => 100101000 => ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [3,2,5] => 1001010000 => ? = 7
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [6,4] => 1000001000 => ? = 2
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [6,4] => 1000001000 => ? = 2
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [[1,6,9,10],[2,8],[3],[4],[5],[7]]
=> [6,4] => 1000001000 => ? = 2
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [3,2,2,3] => 1001010100 => ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> ?
=> ? => ? => ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> ?
=> ? => ? => ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> ?
=> ? => ? => ? = 7
[[1,3,6,7,8,9],[2,5],[4]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [6,2,1] => 100000101 => ? = 5
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [3,2,4] => 100101000 => ? = 5
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [7,2,1] => 1000000101 => ? = 7
[[1,2,3,4,9,10],[5,6,7,8]]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [6,4] => 1000001000 => ? = 2
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [6,4] => 1000001000 => ? = 2
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [5,2,2,1] => 1000010101 => ? = 5
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,3] => 1001010100 => ? = 5
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,5] => 1001010000 => ? = 7
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> ?
=> ? => ? => ? = 3
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> ?
=> ? => ? => ? = 4
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> ?
=> ? => ? => ? = 3
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> ?
=> ? => ? => ? = 4
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> ?
=> ? => ? => ? = 4
[[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [6,4] => 1000001000 => ? = 2
[[1,2,3,4,9],[5,6],[7,8]]
=> [[1,2,3,8,9],[4,5],[6,7]]
=> [3,2,4] => 100101000 => ? = 5
[[1,2,5],[3,4,8],[6,7,9]]
=> ?
=> ? => ? => ? = 3
[[1,2,3,4,5,10],[6,7],[8,9]]
=> [[1,2,3,8,9,10],[4,5],[6,7]]
=> [3,2,5] => 1001010000 => ? = 7
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000097
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => ([],1)
=> 1
[[1,2]]
=> [2] => ([],2)
=> 1
[[1],[2]]
=> [2] => ([],2)
=> 1
[[1,2,3]]
=> [3] => ([],3)
=> 1
[[1,2],[3]]
=> [3] => ([],3)
=> 1
[[1],[2],[3]]
=> [3] => ([],3)
=> 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [4] => ([],4)
=> 1
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,4],[2],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => ([(1,9),(2,9),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => ([(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => ([(2,11),(3,11),(4,11),(5,11),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => ([(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,8),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => ([(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4,9,10],[5,6,7,8]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => ([(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
Description
The order of the largest clique of the graph.
A clique in a graph G is a subset U⊆V(G) such that any pair of vertices in U are adjacent. I.e. the subgraph induced by U is a complete graph.
Matching statistic: St001581
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => ([],1)
=> 1
[[1,2]]
=> [2] => ([],2)
=> 1
[[1],[2]]
=> [2] => ([],2)
=> 1
[[1,2,3]]
=> [3] => ([],3)
=> 1
[[1,2],[3]]
=> [3] => ([],3)
=> 1
[[1],[2],[3]]
=> [3] => ([],3)
=> 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [4] => ([],4)
=> 1
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1,4],[2,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,4],[2],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> 2
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => ([(1,9),(2,9),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => ([(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => ([(2,11),(3,11),(4,11),(5,11),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => ([(3,11),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,8),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => ([(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4,9,10],[5,6,7,8]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => ([(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => ([(3,11),(4,11),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => ([(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000306
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1,0]
=> 0 = 1 - 1
[[1,2]]
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,4,5,6],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5,6],[3,4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5],[3,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4,5,6],[2],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,4,5],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4,5],[2,6],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[3,5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4,5],[2],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,4],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => ?
=> ? = 7 - 1
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => ?
=> ? = 5 - 1
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => ?
=> ? = 5 - 1
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => ?
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => ?
=> ? = 7 - 1
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,4,9,10],[5,6,7,8]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
Description
The bounce count of a Dyck path.
For a Dyck path D of length 2n, this is the number of points (i,i) for 1≤i<n that are touching points of the [[Mp00099|bounce path]] of D.
Matching statistic: St000386
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1,0]
=> 0 = 1 - 1
[[1,2]]
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,4,5,6],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5,6],[3,4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4,5],[3,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4,5,6],[2],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,4,5],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4,5],[2,6],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,4],[3,5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4,5],[2],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,4],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => ?
=> ? = 7 - 1
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => ?
=> ? = 5 - 1
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => ?
=> ? = 5 - 1
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => ?
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => ?
=> ? = 7 - 1
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,4,9,10],[5,6,7,8]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 1
[[1,2]]
=> [2] => [1,1] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2] => [1,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3,4],[2]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3],[2,4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4,5,6],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3,4]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4,5],[3,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4,5],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3],[5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,4],[5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => [1,2,2,1,1,2,1] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => [1,2,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => [1,1,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => [1,2,2,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => [1,2,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => [1,2,2,1,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => [1,2,1,1,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => [1,2,1,1,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => [1,2,1,1,2,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => [1,2,1,1,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => [1,1,2,1,1,1,1,2,1,1] => ([(0,10),(0,11),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => [1,1,1,2,1,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => [1,1,1,2,1,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => [1,1,1,2,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => [1,1,1,2,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => [1,1,1,2,2,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => [1,1,1,2,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => [1,1,1,1,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => [1,1,1,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => [1,1,1,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => [1,1,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => [1,1,1,1,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => [1,1,1,1,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => [1,1,1,1,1,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => [1,2,2,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => [1,1,2,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => [1,1,2,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => [1,1,1,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => [1,2,2,1,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4,9,10],[5,6,7,8]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => [1,2,2,2,1,1,1] => ([(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [2,2,2,2,2] => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => [1,1,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => [1,1,1,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => [1,2,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number α(G) of G.
Matching statistic: St000786
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 1
[[1,2]]
=> [2] => [1,1] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2] => [1,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3,4],[2]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3],[2,4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1],[2],[3],[4],[5]]
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4,5,6],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2,4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3,4]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4,5],[3,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4,5],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4,6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3],[5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2,5],[3,6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,4],[5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2],[3],[5],[6]]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,5,6,7,8],[4]]
=> [4,4] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2
[[1,3,5,6,7,8],[2,4]]
=> [2,2,4] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3
[[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => [1,2,2,1,1,2,1] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => [1,2,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => [1,1,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [2,2,2,4,2] => [1,2,2,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [2,2,4,4] => [1,2,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [2,2,4,2,2] => [1,2,2,1,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [2,4,4,2] => [1,2,1,1,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [2,4,4,2] => [1,2,1,1,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [2,4,2,4] => [1,2,1,1,2,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => [1,2,1,1,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => [1,1,2,1,1,1,1,2,1,1] => ([(0,10),(0,11),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
=> [4,4,2,2] => [1,1,1,2,1,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,6,7,11],[4,5,8,9,10,12]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,6,7,9],[4,5,8,10,11,12]]
=> [4,4,2,2] => [1,1,1,2,1,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
=> [4,2,4,2] => [1,1,1,2,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,8,9],[4,6,7,10,11,12]]
=> [4,2,4,2] => [1,1,1,2,2,1,1,2,1] => ([(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => [1,1,1,2,2,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,2,2,2,2] => [1,1,1,2,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,3,4,5,8],[6,7,9,10,11,12]]
=> [6,3,3] => [1,1,1,1,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4],[5,6],[7,8],[9]]
=> [5,2,2] => [1,1,1,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10]]
=> [6,2,2] => [1,1,1,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4],[5,6,7,8],[9],[10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [4,2,2,2] => [1,1,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [6,3,3] => [1,1,1,1,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [6,3,3] => [1,1,1,1,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 5
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [6,2,2,2] => [1,1,1,1,1,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[1,3,6,7,8,9],[2,5],[4]]
=> [2,2,5] => [1,2,2,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,2,5,9],[3,4,8],[6,7]]
=> [3,3,3] => [1,1,2,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,4,5,9],[2,7,8],[3],[6]]
=> [3,3,3] => [1,1,2,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[[1,6,9],[2,8],[3],[4],[5],[7]]
=> [5,2,2] => [1,1,1,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,3,6,7,8,9,10],[2,5],[4]]
=> [2,2,6] => [1,2,2,1,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,4,9,10],[5,6,7,8]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1,3,8,9,10],[2,5],[4,7],[6]]
=> [2,2,2,4] => [1,2,2,2,1,1,1] => ([(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [2,2,2,2,2] => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [4,2,2,2] => [1,1,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> [6,2,2] => [1,1,1,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [2,2,4,4] => [1,2,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [4,4,4] => [1,1,1,2,1,1,2,1,1,1] => ([(0,9),(0,10),(0,11),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [3,3,3,3] => [1,1,2,1,2,1,2,1,1] => ([(0,10),(0,11),(1,8),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, [2,2,2] and [3,2,1]. Therefore, the statistic on this graph is 3.
The following 171 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000012The area of a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. St000010The length of the partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001484The number of singletons of an integer partition. St000159The number of distinct parts of the integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000098The chromatic number of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000201The number of leaf nodes in a binary tree. St000481The number of upper covers of a partition in dominance order. St000568The hook number of a binary tree. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000172The Grundy number of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001197The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000087The number of induced subgraphs. St000099The number of valleys of a permutation, including the boundary. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000527The width of the poset. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules S with Ext_A^i(S,A)=0 for all i=0,1,...,g-1 in the corresponding Nakayama algebra A with global dimension g. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000023The number of inner peaks of a permutation. St000024The number of double up and double down steps of a Dyck path. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000523The number of 2-protected nodes of a rooted tree. St000537The cutwidth of a graph. St000632The jump number of the poset. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001323The independence gap of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001743The discrepancy of a graph. St001869The maximum cut size of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between e_i J and e_j J (the radical of the indecomposable projective modules). St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000353The number of inner valleys of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000779The tier of a permutation. St001480The number of simple summands of the module J^2/J^3. St001638The book thickness of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000035The number of left outer peaks of a permutation. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral.
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