- FindStat

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000110
(load all 2 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [.,.]
 => [1] => 1
[[],[]]
 => [[.,.],.]
 => [1,2] => 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 3
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 6
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 4
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 6
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 12
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 8
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 3
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 6
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 4
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 12
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 8
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 12
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 24
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => 5
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => 4
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => 10
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => 20
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => 3
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => 15
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => 6
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => 12
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => 10
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 30
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => 20
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => 30
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => 60
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 10
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 8
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 20
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 40
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 3
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 6
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 15
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 30
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 4
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => 12
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 8
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 12
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 24

Description

The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.

Matching statistic: St000063

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 54% ●values known / values provided: 75%●distinct values known / distinct values provided: 54%

Values

[[]]
 => [.,.]
 => [1,0]
 => []
 => 1
[[],[]]
 => [[.,.],.]
 => [1,1,0,0]
 => []
 => 1
[[[]]]
 => [.,[.,.]]
 => [1,0,1,0]
 => [1]
 => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => []
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [2]
 => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1]
 => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 3
[[[[]]]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1]
 => 6
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => []
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 4
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 6
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 12
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 8
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 3
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 6
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 4
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 12
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 8
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 12
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 24
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 5
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 4
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 10
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 20
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 3
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 15
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 6
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 12
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 10
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 30
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 20
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 30
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 60
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 10
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 8
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 20
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 40
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 3
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 6
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 15
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 30
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 4
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 12
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 8
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 12
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 24
[[],[],[],[[],[[]]]]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [6,4,4]
 => ? = 105
[[],[],[],[[[[]]]]]
 => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => ? = 210
[[],[],[[],[],[[]]]]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3]
 => ? = 140
[[],[],[[],[[]],[]]]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3]
 => ? = 105
[[],[],[[],[[],[]]]]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3]
 => ? = 210
[[],[],[[],[[[]]]]]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3]
 => ? = 420
[[],[],[[[]],[],[]]]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3]
 => ? = 70
[[],[],[[[[[]]]]]]
 => [[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3]
 => ? = 840
[[],[[],[]],[[],[]]]
 => [[[.,.],[[.,.],.]],[[.,.],.]]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2]
 => ? = 126
[[],[[],[[],[[]]]]]
 => [[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2]
 => ? = 630
[[],[[],[[[]],[]]]]
 => [[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2]
 => ? = 420
[[],[[],[[[[]]]]]]
 => [[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2]
 => ? = 1260
[[],[[[[[[]]]]]]]
 => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => ? = 2520
[[[[[[[[]]]]]]]]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => ? = 5040
[[],[],[],[],[],[[[]]]]
 => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [7,6]
 => ? = 56
[[],[],[],[],[[],[],[]]]
 => [[[[[.,.],.],.],.],[[[.,.],.],.]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [5,5,5]
 => ? = 56
[[],[],[],[],[[],[[]]]]
 => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [7,5,5]
 => ? = 168
[[],[],[],[],[[[[]]]]]
 => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5]
 => ? = 336
[[],[],[],[[],[],[],[]]]
 => [[[[.,.],.],.],[[[[.,.],.],.],.]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4]
 => ? = 70
[[],[],[],[[],[],[[]]]]
 => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4]
 => ? = 280
[[],[],[],[[],[[],[]]]]
 => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4]
 => ? = 420
[[],[],[],[[],[[[]]]]]
 => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4]
 => ? = 840
[[],[],[],[[[[[]]]]]]
 => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4]
 => ? = 1680
[[],[],[[[]]],[[],[]]]
 => [[[[.,.],.],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
 => [6,6,4,3]
 => ? = 560
[[],[],[[],[],[[],[]]]]
 => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,6,3,3,3]
 => ? = 560
[[],[],[[],[],[[[]]]]]
 => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3]
 => ? = 1120
[[],[],[[],[[],[]],[]]]
 => [[[.,.],.],[[[.,.],[[.,.],.]],.]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [5,5,3,3,3]
 => ? = 336
[[],[],[[],[[],[[]]]]]
 => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3]
 => ? = 1680
[[],[],[[],[[[[]]]]]]
 => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3]
 => ? = 3360
[[],[],[[[],[],[[]]]]]
 => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3]
 => ? = 1120
[[],[],[[[[],[],[]]]]]
 => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3]
 => ? = 1120
[[],[],[[[[[[]]]]]]]
 => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3]
 => ? = 6720
[[],[[]],[[[],[]],[]]]
 => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [5,5,4,4,2]
 => ? = 630
[[],[[],[[],[]]],[[]]]
 => [[[.,.],[[.,.],[[.,.],.]]],[.,.]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [7,4,4,2,2]
 => ? = 720
[[],[[[[[]]]]],[[]]]
 => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2]
 => ? = 2880
[[],[[],[],[],[[],[]]]]
 => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,6,2,2,2,2]
 => ? = 420
[[],[[],[[],[],[],[]]]]
 => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2,2]
 => ? = 420
[[],[[],[[],[[],[]]]]]
 => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,2,2]
 => ? = 2520
[[],[[],[[],[[[]]]]]]
 => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2]
 => ? = 5040
[[],[[],[[[[[]]]]]]]
 => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2]
 => ? = 10080
[[],[[[],[[[[]]]]]]]
 => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2]
 => ? = 10080
[[],[[[[],[[[]]]]]]]
 => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2]
 => ? = 10080
[[],[[[[[],[[]]]]]]]
 => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2]
 => ? = 10080
[[],[[[[[[],[]]]]]]]
 => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,2]
 => ? = 10080
[[],[[[[[[[]]]]]]]]
 => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2]
 => ? = 20160
[[[]],[],[[[]],[],[]]]
 => [[[.,[.,.]],.],[[[.,[.,.]],.],.]]
 => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
 => [5,4,4,4,1]
 => ? = 280
[[[]],[[]],[[]],[[]]]
 => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => [7,5,3,1]
 => ? = 384
[[[]],[[]],[[[[]]]]]
 => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,1]
 => ? = 2688
[[[]],[[[]]],[[],[]]]
 => [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
 => [6,6,4,3,1]
 => ? = 1120
[[[]],[[[[]]]],[[]]]
 => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,1]
 => ? = 1920

Description

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition

$\lambda$ also counts cover-inclusive Dyck tilings of

$\lambda\setminus\mu$ , summed over all

$\mu$ , as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

Matching statistic: St000100
(load all 2 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 67%●distinct values known / distinct values provided: 53%

Values

[[]]
 => [.,.]
 => [1] => ([],1)
 => ? = 1
[[],[]]
 => [[.,.],.]
 => [1,2] => ([(0,1)],2)
 => 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => ([],2)
 => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(1,2)],3)
 => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(1,2)],3)
 => 3
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([],3)
 => 6
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 4
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(2,3)],4)
 => 12
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 8
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 6
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 4
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(2,3)],4)
 => 12
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 8
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(2,3)],4)
 => 12
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([],4)
 => 24
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 5
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 10
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 20
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 3
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 15
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 6
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 12
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 10
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 30
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 20
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 30
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 60
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 10
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 8
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => 20
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 40
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 3
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 6
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 15
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 30
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 12
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 8
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 12
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 24
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 5
[[],[],[],[],[],[[]]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 7
[[],[],[],[],[[],[]]]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 21
[[],[],[],[],[[[]]]]
 => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 42
[[],[],[],[[]],[[]]]
 => [[[[[.,.],.],.],[.,.]],[.,.]]
 => [7,5,1,2,3,4,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[[],[],[],[[],[],[]]]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 35
[[],[],[],[[],[[]]]]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 105
[[],[],[],[[[[]]]]]
 => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => ? = 210
[[],[],[[]],[],[[]]]
 => [[[[[.,.],.],[.,.]],.],[.,.]]
 => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[[],[],[[],[],[[]]]]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 140
[[],[],[[],[[]],[]]]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
 => ? = 105
[[],[],[[],[[],[]]]]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 210
[[],[],[[],[[[]]]]]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 420
[[],[],[[[]],[],[]]]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [5,4,6,7,1,2,3] => ([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
 => ? = 70
[[],[],[[[[[]]]]]]
 => [[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => ? = 840
[[],[[]],[],[],[[]]]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[[],[[],[]],[],[[]]]
 => [[[[.,.],[[.,.],.]],.],[.,.]]
 => [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? = 42
[[],[[],[]],[[],[]]]
 => [[[.,.],[[.,.],.]],[[.,.],.]]
 => [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
 => ? = 126
[[],[[],[],[],[],[]]]
 => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 21
[[],[[],[[],[[]]]]]
 => [[.,.],[[.,.],[[.,.],[.,.]]]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 630
[[],[[],[[[]],[]]]]
 => [[.,.],[[.,.],[[.,[.,.]],.]]]
 => [6,5,7,3,4,1,2] => ([(0,6),(1,6),(2,5),(3,4)],7)
 => ? = 420
[[],[[],[[[[]]]]]]
 => [[.,.],[[.,.],[.,[.,[.,.]]]]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => ? = 1260
[[],[[[[[[]]]]]]]
 => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => ? = 2520
[[[]],[],[],[],[[]]]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => [7,2,1,3,4,5,6] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
 => ? = 14
[[[]],[[]],[],[[]]]
 => [[[[.,[.,.]],[.,.]],.],[.,.]]
 => [7,4,2,1,3,5,6] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[[[],[]],[],[],[[]]]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => [7,2,3,1,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[[[[]]],[],[],[[]]]
 => [[[[.,[.,[.,.]]],.],.],[.,.]]
 => [7,3,2,1,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
 => ? = 42
[[[],[],[]],[],[[]]]
 => [[[.,[[[.,.],.],.]],.],[.,.]]
 => [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[[[[]],[]],[],[[]]]
 => [[[.,[[.,[.,.]],.]],.],[.,.]]
 => [7,3,2,4,1,5,6] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[[[],[],[],[]],[[]]]
 => [[.,[[[[.,.],.],.],.]],[.,.]]
 => [7,2,3,4,5,1,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[[[],[],[],[],[],[]]]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 7
[[[],[],[],[[]],[]]]
 => [.,[[[[[.,.],.],.],[.,.]],.]]
 => [6,2,3,4,5,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[[[],[],[[]],[],[]]]
 => [.,[[[[[.,.],.],[.,.]],.],.]]
 => [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[[[],[[]],[],[],[]]]
 => [.,[[[[[.,.],[.,.]],.],.],.]]
 => [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[[[],[[],[]],[],[]]]
 => [.,[[[[.,.],[[.,.],.]],.],.]]
 => [4,5,2,3,6,7,1] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? = 42
[[[[]],[],[],[],[]]]
 => [.,[[[[[.,[.,.]],.],.],.],.]]
 => [3,2,4,5,6,7,1] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
 => ? = 14
[[[[]],[[]],[],[]]]
 => [.,[[[[.,[.,.]],[.,.]],.],.]]
 => [5,3,2,4,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[[[[],[]],[],[],[]]]
 => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[[[[[]]],[],[],[]]]
 => [.,[[[[.,[.,[.,.]]],.],.],.]]
 => [4,3,2,5,6,7,1] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
 => ? = 42
[[[[],[],[]],[],[]]]
 => [.,[[[.,[[[.,.],.],.]],.],.]]
 => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[[[[[]],[]],[],[]]]
 => [.,[[[.,[[.,[.,.]],.]],.],.]]
 => [4,3,5,2,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[[[[],[],[],[]],[]]]
 => [.,[[.,[[[[.,.],.],.],.]],.]]
 => [3,4,5,6,2,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[[[[[[[[]]]]]]]]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ([],7)
 => ? = 5040
[[],[],[],[],[],[],[[]]]
 => [[[[[[[.,.],.],.],.],.],.],[.,.]]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 8
[[],[],[],[],[],[[]],[]]
 => [[[[[[[.,.],.],.],.],.],[.,.]],.]
 => [7,1,2,3,4,5,6,8] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 7
[[],[],[],[],[],[[],[]]]
 => [[[[[[.,.],.],.],.],.],[[.,.],.]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 28
[[],[],[],[],[],[[[]]]]
 => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
 => ? = 56
[[],[],[],[],[[]],[],[]]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => [6,1,2,3,4,5,7,8] => ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
 => ? = 6
[[],[],[],[],[[],[]],[]]
 => [[[[[[.,.],.],.],.],[[.,.],.]],.]
 => [6,7,1,2,3,4,5,8] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 21
[[],[],[],[],[[],[],[]]]
 => [[[[[.,.],.],.],.],[[[.,.],.],.]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? = 56

Description

The number of linear extensions of a poset.

Matching statistic: St000085
(load all 2 compositions to match this statistic)

Mapping

Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 56% ●values known / values provided: 63%●distinct values known / distinct values provided: 56%

Values

[[]]
 => [[]]
 => [[]]
 => 1
[[],[]]
 => [[],[]]
 => [[[]]]
 => 1
[[[]]]
 => [[[]]]
 => [[],[]]
 => 2
[[],[],[]]
 => [[],[],[]]
 => [[[[]]]]
 => 1
[[],[[]]]
 => [[[]],[]]
 => [[],[[]]]
 => 3
[[[]],[]]
 => [[],[[]]]
 => [[[],[]]]
 => 2
[[[],[]]]
 => [[[],[]]]
 => [[[]],[]]
 => 3
[[[[]]]]
 => [[[[]]]]
 => [[],[],[]]
 => 6
[[],[],[],[]]
 => [[],[],[],[]]
 => [[[[[]]]]]
 => 1
[[],[],[[]]]
 => [[[]],[],[]]
 => [[],[[[]]]]
 => 4
[[],[[]],[]]
 => [[],[[]],[]]
 => [[[],[[]]]]
 => 3
[[],[[],[]]]
 => [[[],[]],[]]
 => [[[]],[[]]]
 => 6
[[],[[[]]]]
 => [[[[]]],[]]
 => [[],[],[[]]]
 => 12
[[[]],[],[]]
 => [[],[],[[]]]
 => [[[[],[]]]]
 => 2
[[[]],[[]]]
 => [[[]],[[]]]
 => [[],[[],[]]]
 => 8
[[[],[]],[]]
 => [[],[[],[]]]
 => [[[[]],[]]]
 => 3
[[[[]]],[]]
 => [[],[[[]]]]
 => [[[],[],[]]]
 => 6
[[[],[],[]]]
 => [[[],[],[]]]
 => [[[[]]],[]]
 => 4
[[[],[[]]]]
 => [[[[]],[]]]
 => [[],[[]],[]]
 => 12
[[[[]],[]]]
 => [[[],[[]]]]
 => [[[],[]],[]]
 => 8
[[[[],[]]]]
 => [[[[],[]]]]
 => [[[]],[],[]]
 => 12
[[[[[]]]]]
 => [[[[[]]]]]
 => [[],[],[],[]]
 => 24
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [[[[[[]]]]]]
 => 1
[[],[],[],[[]]]
 => [[[]],[],[],[]]
 => [[],[[[[]]]]]
 => 5
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [[[],[[[]]]]]
 => 4
[[],[],[[],[]]]
 => [[[],[]],[],[]]
 => [[[]],[[[]]]]
 => 10
[[],[],[[[]]]]
 => [[[[]]],[],[]]
 => [[],[],[[[]]]]
 => 20
[[],[[]],[],[]]
 => [[],[],[[]],[]]
 => [[[[],[[]]]]]
 => 3
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [[],[[],[[]]]]
 => 15
[[],[[],[]],[]]
 => [[],[[],[]],[]]
 => [[[[]],[[]]]]
 => 6
[[],[[[]]],[]]
 => [[],[[[]]],[]]
 => [[[],[],[[]]]]
 => 12
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [[[[]]],[[]]]
 => 10
[[],[[],[[]]]]
 => [[[[]],[]],[]]
 => [[],[[]],[[]]]
 => 30
[[],[[[]],[]]]
 => [[[],[[]]],[]]
 => [[[],[]],[[]]]
 => 20
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [[[]],[],[[]]]
 => 30
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [[],[],[],[[]]]
 => 60
[[[]],[],[],[]]
 => [[],[],[],[[]]]
 => [[[[[],[]]]]]
 => 2
[[[]],[],[[]]]
 => [[[]],[],[[]]]
 => [[],[[[],[]]]]
 => 10
[[[]],[[]],[]]
 => [[],[[]],[[]]]
 => [[[],[[],[]]]]
 => 8
[[[]],[[],[]]]
 => [[[],[]],[[]]]
 => [[[]],[[],[]]]
 => 20
[[[]],[[[]]]]
 => [[[[]]],[[]]]
 => [[],[],[[],[]]]
 => 40
[[[],[]],[],[]]
 => [[],[],[[],[]]]
 => [[[[[]],[]]]]
 => 3
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => [[[[],[],[]]]]
 => 6
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => [[],[[[]],[]]]
 => 15
[[[[]]],[[]]]
 => [[[]],[[[]]]]
 => [[],[[],[],[]]]
 => 30
[[[],[],[]],[]]
 => [[],[[],[],[]]]
 => [[[[[]]],[]]]
 => 4
[[[],[[]]],[]]
 => [[],[[[]],[]]]
 => [[[],[[]],[]]]
 => 12
[[[[]],[]],[]]
 => [[],[[],[[]]]]
 => [[[[],[]],[]]]
 => 8
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [[[[]],[],[]]]
 => 12
[[[[[]]]],[]]
 => [[],[[[[]]]]]
 => [[[],[],[],[]]]
 => 24
[[],[],[],[],[],[],[]]
 => [[],[],[],[],[],[],[]]
 => [[[[[[[[]]]]]]]]
 => ? = 1
[[],[],[],[],[[]],[]]
 => [[],[[]],[],[],[],[]]
 => [[[],[[[[[]]]]]]]
 => ? = 6
[[],[],[],[],[[[]]]]
 => [[[[]]],[],[],[],[]]
 => [[],[],[[[[[]]]]]]
 => ? = 42
[[],[],[],[[]],[],[]]
 => [[],[],[[]],[],[],[]]
 => [[[[],[[[[]]]]]]]
 => ? = 5
[[],[],[],[[],[]],[]]
 => [[],[[],[]],[],[],[]]
 => [[[[]],[[[[]]]]]]
 => ? = 15
[[],[],[],[[],[],[]]]
 => [[[],[],[]],[],[],[]]
 => [[[[]]],[[[[]]]]]
 => ? = 35
[[],[],[],[[],[[]]]]
 => [[[[]],[]],[],[],[]]
 => [[],[[]],[[[[]]]]]
 => ? = 105
[[],[],[],[[[[]]]]]
 => [[[[[]]]],[],[],[]]
 => [[],[],[],[[[[]]]]]
 => ? = 210
[[],[],[[]],[],[],[]]
 => [[],[],[],[[]],[],[]]
 => [[[[[],[[[]]]]]]]
 => ? = 4
[[],[],[[],[],[]],[]]
 => [[],[[],[],[]],[],[]]
 => [[[[[]]],[[[]]]]]
 => ? = 20
[[],[],[[],[],[[]]]]
 => [[[[]],[],[]],[],[]]
 => [[],[[[]]],[[[]]]]
 => ? = 140
[[],[],[[],[[]],[]]]
 => [[[],[[]],[]],[],[]]
 => [[[],[[]]],[[[]]]]
 => ? = 105
[[],[],[[],[[],[]]]]
 => [[[[],[]],[]],[],[]]
 => [[[]],[[]],[[[]]]]
 => ? = 210
[[],[],[[],[[[]]]]]
 => [[[[[]]],[]],[],[]]
 => [[],[],[[]],[[[]]]]
 => ? = 420
[[],[],[[[]],[],[]]]
 => [[[],[],[[]]],[],[]]
 => [[[[],[]]],[[[]]]]
 => ? = 70
[[],[],[[[[[]]]]]]
 => [[[[[[]]]]],[],[]]
 => [[],[],[],[],[[[]]]]
 => ? = 840
[[],[[]],[],[],[],[]]
 => [[],[],[],[],[[]],[]]
 => [[[[[[],[[]]]]]]]
 => ? = 3
[[],[[],[]],[],[],[]]
 => [[],[],[],[[],[]],[]]
 => [[[[[[]],[[]]]]]]
 => ? = 6
[[],[[],[]],[[],[]]]
 => [[[],[]],[[],[]],[]]
 => [[[]],[[[]],[[]]]]
 => ? = 126
[[],[[],[[],[]]],[]]
 => [[],[[[],[]],[]],[]]
 => [[[[]],[[]],[[]]]]
 => ? = 90
[[],[[],[[],[[]]]]]
 => [[[[[]],[]],[]],[]]
 => [[],[[]],[[]],[[]]]
 => ? = 630
[[],[[],[[[]],[]]]]
 => [[[[],[[]]],[]],[]]
 => [[[],[]],[[]],[[]]]
 => ? = 420
[[],[[],[[[[]]]]]]
 => [[[[[[]]]],[]],[]]
 => [[],[],[],[[]],[[]]]
 => ? = 1260
[[],[[[[[[]]]]]]]
 => [[[[[[[]]]]]],[]]
 => [[],[],[],[],[],[[]]]
 => ? = 2520
[[[]],[],[],[],[],[]]
 => [[],[],[],[],[],[[]]]
 => [[[[[[[],[]]]]]]]
 => ? = 2
[[[],[]],[],[],[],[]]
 => [[],[],[],[],[[],[]]]
 => [[[[[[[]],[]]]]]]
 => ? = 3
[[[[]]],[],[],[],[]]
 => [[],[],[],[],[[[]]]]
 => [[[[[[],[],[]]]]]]
 => ? = 6
[[[],[],[]],[],[],[]]
 => [[],[],[],[[],[],[]]]
 => [[[[[[[]]],[]]]]]
 => ? = 4
[[[[[]]]],[],[],[]]
 => [[],[],[],[[[[]]]]]
 => [[[[[],[],[],[]]]]]
 => ? = 24
[[[],[],[],[]],[],[]]
 => [[],[],[[],[],[],[]]]
 => [[[[[[[]]]],[]]]]
 => ? = 5
[[[[[[]]]]],[],[]]
 => [[],[],[[[[[]]]]]]
 => [[[[],[],[],[],[]]]]
 => ? = 120
[[[],[],[],[],[]],[]]
 => [[],[[],[],[],[],[]]]
 => [[[[[[[]]]]],[]]]
 => ? = 6
[[[[],[]],[[]]],[]]
 => [[],[[[]],[[],[]]]]
 => [[[],[[[]],[]],[]]]
 => ? = 90
[[[[[]]],[[]]],[]]
 => [[],[[[]],[[[]]]]]
 => [[[],[[],[],[]],[]]]
 => ? = 180
[[[[[[]]]],[]],[]]
 => [[],[[],[[[[]]]]]]
 => [[[[],[],[],[]],[]]]
 => ? = 144
[[[[],[],[[]]]],[]]
 => [[],[[[[]],[],[]]]]
 => [[[],[[[]]],[],[]]]
 => ? = 120
[[[[],[[[]]]]],[]]
 => [[],[[[[[]]],[]]]]
 => [[[],[],[[]],[],[]]]
 => ? = 360
[[[[[]],[[]]]],[]]
 => [[],[[[[]],[[]]]]]
 => [[[],[[],[]],[],[]]]
 => ? = 240
[[[[[],[]],[]]],[]]
 => [[],[[[],[[],[]]]]]
 => [[[[[]],[]],[],[]]]
 => ? = 90
[[[[[[]]],[]]],[]]
 => [[],[[[],[[[]]]]]]
 => [[[[],[],[]],[],[]]]
 => ? = 180
[[[[[],[],[]]]],[]]
 => [[],[[[[],[],[]]]]]
 => [[[[[]]],[],[],[]]]
 => ? = 120
[[[[[[]],[]]]],[]]
 => [[],[[[[],[[]]]]]]
 => [[[[],[]],[],[],[]]]
 => ? = 240
[[[[[[],[]]]]],[]]
 => [[],[[[[[],[]]]]]]
 => [[[[]],[],[],[],[]]]
 => ? = 360
[[[[[[[]]]]]],[]]
 => [[],[[[[[[]]]]]]]
 => [[[],[],[],[],[],[]]]
 => ? = 720
[[[[[[[[]]]]]]]]
 => [[[[[[[[]]]]]]]]
 => [[],[],[],[],[],[],[]]
 => ? = 5040
[[],[],[],[],[],[],[],[]]
 => [[],[],[],[],[],[],[],[]]
 => [[[[[[[[[]]]]]]]]]
 => ? = 1
[[],[],[],[],[],[[]],[]]
 => [[],[[]],[],[],[],[],[]]
 => [[[],[[[[[[]]]]]]]]
 => ? = 7
[[],[],[],[],[],[[],[]]]
 => [[[],[]],[],[],[],[],[]]
 => [[[]],[[[[[[]]]]]]]
 => ? = 28
[[],[],[],[],[],[[[]]]]
 => [[[[]]],[],[],[],[],[]]
 => [[],[],[[[[[[]]]]]]]
 => ? = 56
[[],[],[],[],[[]],[],[]]
 => [[],[],[[]],[],[],[],[]]
 => [[[[],[[[[[]]]]]]]]
 => ? = 6

Description

The number of linear extensions of the tree. We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree

$T$ on

$n$ vertices, the number of linear extensions is

$\frac{n!}{\prod_{v\in T}|T_v|},$ where

$T_v$ is the number of vertices of the subtree rooted at

$v$ .

Matching statistic: St001855

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%

Values

[[]]
 => [.,.]
 => [1] => [1] => 1
[[],[]]
 => [[.,.],.]
 => [1,2] => [1,2] => 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [2,1] => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => 3
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => 3
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 6
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,2,3] => 4
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2,4] => 3
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,4,1,2] => 6
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,3,1,2] => 12
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [4,2,1,3] => 8
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => 3
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 6
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => 4
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,2,3,1] => 12
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => 8
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => 12
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 24
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 5
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [4,1,2,3,5] => ? = 4
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,5,1,2,3] => ? = 10
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [5,4,1,2,3] => ? = 20
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 3
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [5,3,1,2,4] => ? = 15
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [3,4,1,2,5] => ? = 6
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 12
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [3,4,5,1,2] => ? = 10
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,3,4,1,2] => ? = 30
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [4,3,5,1,2] => ? = 20
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,5,3,1,2] => ? = 30
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 60
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [5,2,1,3,4] => ? = 10
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [4,2,1,3,5] => ? = 8
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,5,2,1,3] => ? = 20
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,4,2,1,3] => ? = 40
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? = 6
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [5,2,3,1,4] => ? = 15
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [5,3,2,1,4] => ? = 30
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [2,3,4,1,5] => ? = 4
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [4,2,3,1,5] => ? = 12
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,2,4,1,5] => ? = 8
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,4,2,1,5] => ? = 12
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 24
[[[],[],[],[]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 5
[[[],[],[[]]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,2,3,4,1] => ? = 20
[[[],[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [4,2,3,5,1] => ? = 15
[[[],[[],[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [4,5,2,3,1] => ? = 30
[[[],[[[]]]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,4,2,3,1] => ? = 60
[[[[]],[],[]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => ? = 10
[[[[]],[[]]]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,3,2,4,1] => ? = 40
[[[[],[]],[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => ? = 15
[[[[[]]],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => ? = 30
[[[[],[],[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => ? = 20
[[[[],[[]]]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,3,4,2,1] => ? = 60
[[[[[]],[]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => ? = 40
[[[[[],[]]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => ? = 60
[[[[[[]]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 120
[[],[],[],[],[],[]]
 => [[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1
[[],[],[],[],[[]]]
 => [[[[[.,.],.],.],.],[.,.]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 6
[[],[],[],[[]],[]]
 => [[[[[.,.],.],.],[.,.]],.]
 => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 5
[[],[],[],[[],[]]]
 => [[[[.,.],.],.],[[.,.],.]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 15
[[],[],[],[[[]]]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 30
[[],[],[[]],[],[]]
 => [[[[[.,.],.],[.,.]],.],.]
 => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 4
[[],[],[[]],[[]]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => [6,4,1,2,3,5] => [6,4,1,2,3,5] => ? = 24
[[],[],[[],[]],[]]
 => [[[[.,.],.],[[.,.],.]],.]
 => [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 10
[[],[],[[[]]],[]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => [5,4,1,2,3,6] => [5,4,1,2,3,6] => ? = 20

Description

The number of signed permutations less than or equal to a signed permutation in left weak order.