Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000733
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Mapping
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => 1
[[1,2]]
 => [[1,2]]
 => 1
[[1],[2]]
 => [[1],[2]]
 => 2
[[1,2,3]]
 => [[1,2,3]]
 => 1
[[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[1,2],[3]]
 => [[1,3],[2]]
 => 1
[[1],[2],[3]]
 => [[1],[2],[3]]
 => 3
[[1,2,3,4]]
 => [[1,2,3,4]]
 => 1
[[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 2
[[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 1
[[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[[1,2],[3,4]]
 => [[1,2],[3,4]]
 => 2
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 3
[[1,3],[2],[4]]
 => [[1,3],[2],[4]]
 => 3
[[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 1
[[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => 4
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 1
[[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => 2
[[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 1
[[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => 2
[[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => 2
[[1,3,4],[2,5]]
 => [[1,3,4],[2,5]]
 => 2
[[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 1
[[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => 1
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 3
[[1,2,5],[3],[4]]
 => [[1,2,5],[3],[4]]
 => 1
[[1,3,4],[2],[5]]
 => [[1,3,4],[2],[5]]
 => 3
[[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 1
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[[1,4],[2,5],[3]]
 => [[1,3],[2,4],[5]]
 => 3
[[1,3],[2,5],[4]]
 => [[1,2],[3,4],[5]]
 => 3
[[1,2],[3,5],[4]]
 => [[1,2],[3,5],[4]]
 => 2
[[1,3],[2,4],[5]]
 => [[1,4],[2,5],[3]]
 => 2
[[1,2],[3,4],[5]]
 => [[1,3],[2,5],[4]]
 => 2
[[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => 4
[[1,4],[2],[3],[5]]
 => [[1,3],[2],[4],[5]]
 => 4
[[1,3],[2],[4],[5]]
 => [[1,4],[2],[3],[5]]
 => 4
[[1,2],[3],[4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => 5
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => 1
[[1,3,4,5,6],[2]]
 => [[1,2,3,4,5],[6]]
 => 2
[[1,2,4,5,6],[3]]
 => [[1,2,3,4,6],[5]]
 => 1
[[1,2,3,5,6],[4]]
 => [[1,2,3,5,6],[4]]
 => 1
[[1,2,3,4,6],[5]]
 => [[1,2,4,5,6],[3]]
 => 1
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[[1,3,5,6],[2,4]]
 => [[1,2,3,5],[4,6]]
 => 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001390
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [1,2] => 1
[[1],[2]]
 => [2,1] => [2,1] => 2
[[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => 2
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 3
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,3,4,1] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3,4,2] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,4,2,1] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 4
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,3,4,5,1] => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,4,5,2] => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,5,3] => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,4,5,1,3] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,5,1,2] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3,5,1,4] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,3,5,2,4] => 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,4,5,2,3] => 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,5,3,1] => 3
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,5,3,2] => 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,3,5,4,1] => 3
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,4,2] => 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,5,2,4,1] => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,5,2,3,1] => 3
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,1,3,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,5,1,4,3] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [3,5,1,4,2] => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,5,3,2,1] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,5,4,2,1] => 4
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,3,1] => 4
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 5
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,3,4,5,6,1] => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,3,4,5,6,2] => 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,4,5,6,3] => 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,5,6,4] => 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [2,4,5,6,1,3] => 2
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
Matching statistic: St001232
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [1,0]
 => [1,0]
 => 0 = 1 - 1
[[1,2]]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 3 - 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[1,2,4],[3]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,3],[2,4]]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[1,2],[3,4]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 3 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 4 - 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 2 - 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 1 - 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 3 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 4 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4 - 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 5 - 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 - 1
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 1 - 1
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 - 1
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1 - 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
Mapping
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => ([],1)
 => ([(0,1)],2)
 => 2 = 1 + 1
[[1,2]]
 => [2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1],[2]]
 => [1,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,2,3]]
 => [3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[[1,2],[3]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,2,3,4]]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2,3],[4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,3],[2,4]]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[[1,3],[2],[4]]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[[1,2],[3],[4]]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1],[2],[3],[4]]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,2,4,5],[3]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,3,5],[2,4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,3,4],[2,5]]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,2,4],[3,5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[[1,3,5],[2],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,3,4],[2],[5]]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[[1,2,4],[3],[5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,2,3],[4],[5]]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,4],[2,5],[3]]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3 + 1
[[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[[1,2],[3,5],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,3],[2,4],[5]]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 2 + 1
[[1,2],[3,4],[5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,5],[2],[3],[4]]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[[1,4],[2],[3],[5]]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 4 + 1
[[1,3],[2],[4],[5]]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ? = 4 + 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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
[[1],[2],[3],[4],[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)
 => ([(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)
 => 6 = 5 + 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[1,2,4,5,6],[3]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,2,3,5,6],[4]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,2,3,4,5],[6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,3,5,6],[2,4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[1,3,4,6],[2,5]]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,2,3,6],[4,5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,3,4,5],[2,6]]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,2,3,5],[4,6]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[[1,3,5,6],[2],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[[1],[2],[3],[4],[5],[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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.