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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001214
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> ([],1)
=> [1]
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 0
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St000624
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 80%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [2,1] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[2,1,3] => [2,3,1] => [1,3,2] => [2,3,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [3,1,2] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [4,1,3,2] => [2,3,1,4] => 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 0
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 0
[2,1,3,4] => [3,4,2,1] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[2,3,1,4] => [3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [3,4,1,2] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 0
[3,1,2,4] => [2,4,3,1] => [1,4,3,2] => [2,3,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [2,3,4,1] => [1,2,4,3] => [3,4,2,1] => 1
[3,2,4,1] => [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[4,2,1,3] => [1,3,4,2] => [1,2,4,3] => [3,4,2,1] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => [2,1,3,4,5] => 0
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,1,3,2] => [2,3,1,4,5] => 1
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => [3,1,2,4,5] => 0
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => [2,3,1,4,5] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => [3,2,1,4,5] => 0
[1,3,2,4,5] => [5,3,4,2,1] => [5,1,4,3,2] => [2,3,4,1,5] => 1
[1,3,2,5,4] => [5,3,4,1,2] => [5,2,4,1,3] => [3,1,4,2,5] => 1
[1,3,4,2,5] => [5,3,2,4,1] => [5,2,1,4,3] => [3,4,1,2,5] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => [4,1,2,3,5] => 0
[1,3,5,2,4] => [5,3,1,4,2] => [5,2,1,4,3] => [3,4,1,2,5] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => [4,2,1,3,5] => 0
[1,4,2,3,5] => [5,2,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => [3,1,4,2,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [5,1,3,2,4] => [4,2,3,1,5] => 1
[1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [3,4,1,2,5] => 1
[1,4,5,3,2] => [5,2,1,3,4] => [5,2,1,3,4] => [4,3,1,2,5] => 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => ? = 0
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => [2,3,1,4,5,6,7] => ? = 1
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [3,1,2,4,5,6,7] => ? = 0
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => [2,3,1,4,5,6,7] => ? = 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => ? = 0
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => [2,3,4,1,5,6,7] => ? = 1
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [7,6,5,2,4,1,3] => [3,1,4,2,5,6,7] => ? = 1
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 1
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [4,1,2,3,5,6,7] => ? = 0
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 1
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => [4,2,1,3,5,6,7] => ? = 0
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,6,5,1,4,3,2] => [2,3,4,1,5,6,7] => ? = 1
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [7,6,5,2,4,1,3] => [3,1,4,2,5,6,7] => ? = 1
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 1
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [7,6,5,1,3,2,4] => [4,2,3,1,5,6,7] => ? = 1
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 1
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => [7,6,5,2,1,3,4] => [4,3,1,2,5,6,7] => ? = 0
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => [2,3,4,1,5,6,7] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [7,6,5,1,4,2,3] => [3,2,4,1,5,6,7] => ? = 1
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 1
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [7,6,5,1,3,2,4] => [4,2,3,1,5,6,7] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => ? = 0
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => [2,3,4,5,1,6,7] => ? = 1
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [7,6,2,5,4,1,3] => [3,1,4,5,2,6,7] => ? = 1
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [7,6,2,5,1,4,3] => [3,4,1,5,2,6,7] => ? = 2
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [7,6,3,5,2,1,4] => [4,1,2,5,3,6,7] => ? = 1
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [7,6,2,5,1,4,3] => [3,4,1,5,2,6,7] => ? = 2
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [7,6,3,5,1,2,4] => [4,2,1,5,3,6,7] => ? = 1
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [7,6,3,2,5,1,4] => [4,1,5,2,3,6,7] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [7,6,3,2,1,5,4] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => [5,1,2,3,4,6,7] => ? = 0
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [7,6,3,2,1,5,4] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [7,6,4,3,1,2,5] => [5,2,1,3,4,6,7] => ? = 0
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,6,2,1,5,4,3] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [7,6,3,2,5,1,4] => [4,1,5,2,3,6,7] => ? = 2
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [7,6,3,1,2,5,4] => [4,5,2,1,3,6,7] => ? = 1
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [7,6,4,1,3,2,5] => [5,2,3,1,4,6,7] => ? = 1
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [7,6,3,2,1,5,4] => [4,5,1,2,3,6,7] => ? = 1
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => [7,6,4,2,1,3,5] => [5,3,1,2,4,6,7] => ? = 0
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,2,1,5,4,3] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [7,6,3,1,5,2,4] => [4,2,5,1,3,6,7] => ? = 2
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [7,6,3,1,2,5,4] => [4,5,2,1,3,6,7] => ? = 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [7,6,4,1,3,2,5] => [5,2,3,1,4,6,7] => ? = 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [7,6,3,1,2,5,4] => [4,5,2,1,3,6,7] => ? = 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [7,6,4,1,2,3,5] => [5,3,2,1,4,6,7] => ? = 0
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => [2,3,4,5,1,6,7] => ? = 1
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [7,6,2,5,4,1,3] => [3,1,4,5,2,6,7] => ? = 1
Description
The normalized sum of the minimal distances to a greater element.
Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is
$$
\sum_{i=1}^n \min_d(\pi_{i-1-d}>\pi_i\text{ or }\pi_{i+1+d}>\pi_i)
$$
A closely related statistic appears in [1].
The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently satisfies the functional equation
$$
(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x^3 = 0.
$$
Matching statistic: St000822
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000822: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 80%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000822: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 80%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,5,7,6,3] => [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,6,7,5,3] => [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 0 + 1
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
Description
The Hadwiger number of the graph.
Also known as clique contraction number, this is the size of the largest complete minor.
Matching statistic: St001330
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,3,4,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,3,5,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,4,5,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,3,6,4,5] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,5,3,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,4,6,3,5] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,3,4,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,3,6,4] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,4,3,6] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,4,6,3] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,3,4,5] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,3,5,4] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,4,3,5] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,4,5,3] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,6,5,3,4] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,3,1,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
[2,3,1,4,6,5] => [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 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.
Matching statistic: St001621
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1] => [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1] => [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001878
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1] => [.,.]
=> ([],1)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 1
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001823
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00223: Permutations —runsort⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,4,2] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,4,3] => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,3,1,4] => [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,4,3,5] => [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,2,4,5,3] => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,3,2,4,5] => [3,1,2,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[1,3,2,5,4] => [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[1,3,4,2,5] => [3,4,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[1,3,4,5,2] => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,3,5,2,4] => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[1,3,5,4,2] => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[1,4,3,5,2] => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,4,5,2,3] => [4,1,5,2,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[1,4,5,3,2] => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[1,5,3,2,4] => [5,1,3,2,4] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[1,5,3,4,2] => [3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[2,1,3,5,4] => [2,5,1,3,4] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
[2,1,4,3,5] => [2,4,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[2,1,4,5,3] => [2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[2,1,5,3,4] => [5,2,1,3,4] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 2
[2,1,5,4,3] => [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[2,3,1,5,4] => [2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[2,3,4,1,5] => [2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,4,1,3,5] => [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2
[2,4,1,5,3] => [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[2,4,3,5,1] => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[2,4,5,1,3] => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[2,4,5,3,1] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[2,5,1,3,4] => [2,1,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 2
[2,5,1,4,3] => [2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 2
[2,5,3,1,4] => [2,5,3,1,4] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1
[2,5,4,1,3] => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[2,5,4,3,1] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[3,1,2,5,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
Description
The Stasinski-Voll length of a signed permutation.
The Stasinski-Voll length of a signed permutation $\sigma$ is
$$
L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\},
$$
where $n$ is the size of $\sigma$.
Matching statistic: St001960
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1,0]
=> [2,1] => 0
[1,2] => [2] => [1,1,0,0]
=> [2,3,1] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [3,1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 0
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 0
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? = 1
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? = 2
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? = 1
[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? = 2
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ? = 1
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 2
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 2
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 0
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 2
[2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 2
[2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1
[2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1
[2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 0
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 2
[2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 2
[2,5,3,1,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1
[2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1
[2,5,4,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1
[2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 0
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001582
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0
[2,1,3] => [2,1,3] => [1,3,2] => [2,3,1] => 1
[2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,4] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,4,2] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,4,3,2] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 1
[2,1,4,3] => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[2,3,1,4] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[2,4,3,1] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 1
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 1
[3,4,1,2] => [3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 1
[4,2,1,3] => [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 1
[4,3,1,2] => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,2,4,3,5] => [4,1,2,3,5] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
[1,2,4,5,3] => [4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,3,2,4,5] => [3,1,2,4,5] => [1,2,4,5,3] => [3,5,4,2,1] => ? = 1
[1,3,2,5,4] => [3,5,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
[1,3,4,2,5] => [3,4,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 1
[1,3,4,5,2] => [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,3,5,2,4] => [5,3,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => ? = 1
[1,3,5,4,2] => [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => ? = 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 1
[1,4,3,5,2] => [4,3,5,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
[1,4,5,2,3] => [4,1,5,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
[1,4,5,3,2] => [4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 1
[1,5,3,2,4] => [5,1,3,2,4] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
[1,5,3,4,2] => [3,5,4,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => ? = 1
[2,1,3,5,4] => [2,5,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ? = 1
[2,1,4,3,5] => [2,4,1,3,5] => [1,3,5,2,4] => [4,2,5,3,1] => ? = 2
[2,1,4,5,3] => [2,4,5,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
[2,1,5,3,4] => [5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ? = 2
[2,1,5,4,3] => [5,2,4,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => ? = 2
[2,3,1,5,4] => [2,3,5,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 2
[2,3,4,1,5] => [2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ? = 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[2,4,1,3,5] => [4,2,1,3,5] => [1,3,5,2,4] => [4,2,5,3,1] => ? = 2
[2,4,1,5,3] => [4,2,5,1,3] => [1,3,2,5,4] => [4,5,2,3,1] => ? = 2
[2,4,3,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 1
[2,4,3,5,1] => [4,2,3,5,1] => [1,2,3,5,4] => [4,5,3,2,1] => ? = 1
[2,4,5,1,3] => [4,5,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
[2,4,5,3,1] => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[2,5,1,3,4] => [2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ? = 2
[2,5,1,4,3] => [2,5,4,1,3] => [1,3,2,5,4] => [4,5,2,3,1] => ? = 2
[2,5,3,1,4] => [2,5,3,1,4] => [1,4,2,5,3] => [3,5,2,4,1] => ? = 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => ? = 1
[2,5,4,1,3] => [5,4,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
[2,5,4,3,1] => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ? = 0
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ? = 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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