Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000032
(load all 5 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => 1
[.,[.,.]]
 => [1,1,0,0]
 => 2
[[.,.],.]
 => [1,0,1,0]
 => 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 5
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 3
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 14
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 9
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 7
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 7
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 4
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 5
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 4
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 2
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 5
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 3
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 42
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 28
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 23
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 23
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 14
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 19
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 12
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 16
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 9
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 19
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 12
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => 9
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 9
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 14
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 9
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 7
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 7
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 10
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 6
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 5
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 10
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 6
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 4
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000082
(load all 3 compositions to match this statistic)
Mapping
St000082: Binary trees ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 44%
Values
[.,.]
 => ? = 1
[.,[.,.]]
 => 2
[[.,.],.]
 => 1
[.,[.,[.,.]]]
 => 5
[.,[[.,.],.]]
 => 3
[[.,.],[.,.]]
 => 2
[[.,[.,.]],.]
 => 2
[[[.,.],.],.]
 => 1
[.,[.,[.,[.,.]]]]
 => 14
[.,[.,[[.,.],.]]]
 => 9
[.,[[.,.],[.,.]]]
 => 7
[.,[[.,[.,.]],.]]
 => 7
[.,[[[.,.],.],.]]
 => 4
[[.,.],[.,[.,.]]]
 => 5
[[.,.],[[.,.],.]]
 => 3
[[.,[.,.]],[.,.]]
 => 4
[[[.,.],.],[.,.]]
 => 2
[[.,[.,[.,.]]],.]
 => 5
[[.,[[.,.],.]],.]
 => 3
[[[.,.],[.,.]],.]
 => 2
[[[.,[.,.]],.],.]
 => 2
[[[[.,.],.],.],.]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => 42
[.,[.,[.,[[.,.],.]]]]
 => 28
[.,[.,[[.,.],[.,.]]]]
 => 23
[.,[.,[[.,[.,.]],.]]]
 => 23
[.,[.,[[[.,.],.],.]]]
 => 14
[.,[[.,.],[.,[.,.]]]]
 => 19
[.,[[.,.],[[.,.],.]]]
 => 12
[.,[[.,[.,.]],[.,.]]]
 => 16
[.,[[[.,.],.],[.,.]]]
 => 9
[.,[[.,[.,[.,.]]],.]]
 => 19
[.,[[.,[[.,.],.]],.]]
 => 12
[.,[[[.,.],[.,.]],.]]
 => 9
[.,[[[.,[.,.]],.],.]]
 => 9
[.,[[[[.,.],.],.],.]]
 => 5
[[.,.],[.,[.,[.,.]]]]
 => 14
[[.,.],[.,[[.,.],.]]]
 => 9
[[.,.],[[.,.],[.,.]]]
 => 7
[[.,.],[[.,[.,.]],.]]
 => 7
[[.,.],[[[.,.],.],.]]
 => 4
[[.,[.,.]],[.,[.,.]]]
 => 10
[[.,[.,.]],[[.,.],.]]
 => 6
[[[.,.],.],[.,[.,.]]]
 => 5
[[[.,.],.],[[.,.],.]]
 => 3
[[.,[.,[.,.]]],[.,.]]
 => 10
[[.,[[.,.],.]],[.,.]]
 => 6
[[[.,.],[.,.]],[.,.]]
 => 4
[[[.,[.,.]],.],[.,.]]
 => 4
[[[[.,.],.],.],[.,.]]
 => 2
[[.,[.,[.,[.,.]]]],.]
 => 14
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ? = 429
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ? = 297
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => ? = 255
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => ? = 255
[.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ? = 165
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => ? = 227
[.,[.,[.,[[.,.],[[.,.],.]]]]]
 => ? = 151
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => ? = 199
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => ? = 123
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => ? = 227
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => ? = 151
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => ? = 123
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => ? = 123
[.,[.,[.,[[[[.,.],.],.],.]]]]
 => ? = 75
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => ? = 202
[.,[.,[[.,.],[.,[[.,.],.]]]]]
 => ? = 136
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ? = 113
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => ? = 113
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => ? = 70
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ? = 161
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ? = 104
[.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ? = 95
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => ? = 61
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => ? = 161
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => ? = 104
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ? = 81
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => ? = 81
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => ? = 47
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => ? = 202
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => ? = 136
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => ? = 113
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => ? = 113
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => ? = 70
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => ? = 95
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => ? = 61
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => ? = 81
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => ? = 47
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => ? = 95
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => ? = 61
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => ? = 47
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => ? = 47
[.,[.,[[[[[.,.],.],.],.],.]]]
 => ? = 27
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => ? = 174
[.,[[.,.],[.,[.,[[.,.],.]]]]]
 => ? = 118
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ? = 99
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => ? = 99
[.,[[.,.],[.,[[[.,.],.],.]]]]
 => ? = 62
[.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ? = 85
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => ? = 55
Description
The number of elements smaller than a binary tree in Tamari order.