Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000681
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 3
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 3
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 0
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 1
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001879
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 40%
Values
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 1 + 2
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 1 + 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 2 + 2
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,5,3,1,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 3 + 2
[1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,3,4,6,5] => [.,[[.,[.,[.,[[.,.],.]]]],.]]
 => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [[.,[.,[.,[[.,[.,.]],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],.]]
 => [[.,[.,[.,[[[.,.],.],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,3,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],.]]
 => [[.,[.,[[.,[.,[.,.]]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,3,5,4] => [.,[[.,[[.,[[.,.],.]],.]],.]]
 => [[.,[.,[[.,[[.,.],.]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,5,3,4] => [.,[[.,[[[.,[.,.]],.],.]],.]]
 => [[.,[.,[[[.,[.,.]],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,2,6,5,4,3] => [.,[[.,[[[[.,.],.],.],.]],.]]
 => [[.,[.,[[[[.,.],.],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,3,4,5] => [.,[[[.,[.,[.,[.,.]]]],.],.]]
 => [[.,[[.,[.,[.,[.,.]]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,3,5,4] => [.,[[[.,[.,[[.,.],.]]],.],.]]
 => [[.,[[.,[.,[[.,.],.]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,5,3,4] => [.,[[[.,[[.,[.,.]],.]],.],.]]
 => [[.,[[.,[[.,[.,.]],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,2,5,4,3] => [.,[[[.,[[[.,.],.],.]],.],.]]
 => [[.,[[.,[[[.,.],.],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,2,3,4] => [.,[[[[.,[.,[.,.]]],.],.],.]]
 => [[.,[[[.,[.,[.,.]]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,2,4,3] => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [[.,[[[.,[[.,.],.]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,4,2,3] => [.,[[[[[.,[.,.]],.],.],.],.]]
 => [[.,[[[[.,[.,.]],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => [[.,[[[[[.,.],.],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 4 + 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 40%
Values
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 1 + 3
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 1 + 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 + 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 3
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 3
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 3
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 2 + 3
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 3
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 3
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 3
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 3
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 3
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 3
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,5,3,1,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 3
[1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 3 + 3
[1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,3,4,6,5] => [.,[[.,[.,[.,[[.,.],.]]]],.]]
 => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [[.,[.,[.,[[.,[.,.]],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],.]]
 => [[.,[.,[.,[[[.,.],.],.]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,3,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],.]]
 => [[.,[.,[[.,[.,[.,.]]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,3,5,4] => [.,[[.,[[.,[[.,.],.]],.]],.]]
 => [[.,[.,[[.,[[.,.],.]],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,5,3,4] => [.,[[.,[[[.,[.,.]],.],.]],.]]
 => [[.,[.,[[[.,[.,.]],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,2,6,5,4,3] => [.,[[.,[[[[.,.],.],.],.]],.]]
 => [[.,[.,[[[[.,.],.],.],.]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,3,4,5] => [.,[[[.,[.,[.,[.,.]]]],.],.]]
 => [[.,[[.,[.,[.,[.,.]]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,3,5,4] => [.,[[[.,[.,[[.,.],.]]],.],.]]
 => [[.,[[.,[.,[[.,.],.]]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,5,3,4] => [.,[[[.,[[.,[.,.]],.]],.],.]]
 => [[.,[[.,[[.,[.,.]],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,2,5,4,3] => [.,[[[.,[[[.,.],.],.]],.],.]]
 => [[.,[[.,[[[.,.],.],.]],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,2,3,4] => [.,[[[[.,[.,[.,.]]],.],.],.]]
 => [[.,[[[.,[.,[.,.]]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,2,4,3] => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [[.,[[[.,[[.,.],.]],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,4,2,3] => [.,[[[[[.,[.,.]],.],.],.],.]]
 => [[.,[[[[.,[.,.]],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => [[.,[[[[[.,.],.],.],.],.]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 4 + 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001330
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 40%
Values
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 1 + 2
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1 + 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 2
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 2
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 2
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 2 + 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 2
[1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,5,3,2,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,4,2,3] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 2
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 2
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ? = 2 + 2
[2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 3 + 2
[2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[2,3,4,5,1] => [1,5,4,3,2] => ([(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),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,4,3,1,5] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,4,5,3,1] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[3,2,5,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,6,5,4,3,2] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,6,5,4,3,1] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,6,5,4,1,2] => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 1 + 2
[3,6,5,4,2,1] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[4,3,6,5,1,2] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 1 + 2
[4,3,6,5,2,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(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 = 4 + 2
[2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,7,6,5,4,1,2] => [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[3,7,6,5,4,2,1] => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,7,6,5,1,2,3] => [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => 3 = 1 + 2
[4,7,6,5,1,3,2] => [2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[4,7,6,5,2,1,3] => [3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[4,7,6,5,2,3,1] => [1,3,2,5,6,7,4] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 1 + 2
[4,7,6,5,3,1,2] => [2,1,3,5,6,7,4] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 3 = 1 + 2
[4,7,6,5,3,2,1] => [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[5,4,7,6,1,3,2] => [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 1 + 2
[5,4,7,6,2,1,3] => [3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 1 + 2
[5,4,7,6,2,3,1] => [1,3,2,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 1 + 2
[5,4,7,6,3,1,2] => [2,1,3,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 1 + 2
[5,4,7,6,3,2,1] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
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.