searching the database
Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000734
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [[1]]
=> 1
[[.,.],.]
=> [1,2] => [1] => [[1]]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [[1],[2]]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [[1],[2]]
=> 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [[1,2]]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [[1],[2]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [[1,2]]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [[1,2],[3]]
=> 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [[1,2],[3]]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [[1,3],[2]]
=> 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [[1,2,3]]
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [[1],[2],[3]]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [[1,2],[3]]
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [[1,2],[3]]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [[1,3],[2]]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [[1,2,3]]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000147
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 90%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 90%
Values
[.,[.,.]]
=> [1,1,0,0]
=> []
=> ?
=> ? = 1 - 1
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> []
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 0 = 1 - 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 2 = 3 - 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 2 = 3 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ?
=> ? = 5 - 1
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,1,1,1]
=> ?
=> ? = 6 - 1
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,2]
=> ?
=> ? = 5 - 1
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,2,2]
=> ?
=> ? = 6 - 1
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,2,2,2]
=> ?
=> ? = 5 - 1
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,4,2,2,2]
=> ?
=> ? = 5 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,2,2,1]
=> ?
=> ? = 5 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> [5,4,3,2,2,1]
=> ? = 6 - 1
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3]
=> ?
=> ? = 5 - 1
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3]
=> ?
=> ? = 5 - 1
[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3]
=> ?
=> ? = 6 - 1
[[.,[.,[.,.]]],[[[.,.],.],[.,.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3]
=> ?
=> ? = 6 - 1
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3]
=> ?
=> ? = 4 - 1
[[.,[.,[.,.]]],[[.,[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3]
=> ?
=> ? = 5 - 1
[[.,[.,[.,.]]],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3]
=> ?
=> ? = 5 - 1
[[.,[.,[.,.]]],[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3]
=> ?
=> ? = 6 - 1
[[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3]
=> ?
=> ? = 6 - 1
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ? = 4 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ? = 4 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ?
=> ? = 6 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ?
=> ? = 6 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> [5,3,3,3,2,1]
=> ? = 6 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> ? = 6 - 1
[[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4]
=> ?
=> ? = 6 - 1
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4]
=> ?
=> ? = 5 - 1
[[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4]
=> ?
=> ? = 6 - 1
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,1]
=> ?
=> ? = 5 - 1
[[.,[.,[[.,.],.]]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,1]
=> ?
=> ? = 6 - 1
[[.,[.,[[.,.],.]]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,1]
=> ?
=> ? = 5 - 1
[[.,[.,[[.,.],.]]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1]
=> ?
=> ? = 6 - 1
[[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1]
=> ?
=> ? = 5 - 1
[[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,1,1]
=> ?
=> ? = 5 - 1
[[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,1,1]
=> ?
=> ? = 6 - 1
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,1,1]
=> ?
=> ? = 5 - 1
[[.,[[.,.],[.,.]]],[[[.,.],.],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1]
=> ?
=> ? = 6 - 1
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,2]
=> ?
=> ? = 5 - 1
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,2]
=> ?
=> ? = 6 - 1
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,2]
=> ?
=> ? = 5 - 1
[[.,[[.,[.,.]],.]],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2]
=> ?
=> ? = 6 - 1
Description
The largest part of an integer partition.
Matching statistic: St001300
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Values
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 0 = 1 - 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => ? => ?
=> ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => ? => ?
=> ? = 3 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ? => ?
=> ? = 2 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [8,7,6,3,4,5,2,1] => ? => ?
=> ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [8,7,3,4,5,6,2,1] => ? => ?
=> ? = 4 - 1
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [8,6,4,3,5,7,2,1] => ? => ?
=> ? = 5 - 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => ? => ?
=> ? = 2 - 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4 - 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [7,6,5,4,8,2,3,1] => ? => ?
=> ? = 2 - 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 5 - 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 5 - 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => ? => ?
=> ? = 4 - 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [8,7,6,3,4,2,5,1] => ? => ?
=> ? = 4 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => ? => ?
=> ? = 4 - 1
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [8,7,6,3,2,4,5,1] => ? => ?
=> ? = 4 - 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => ? => ?
=> ? = 4 - 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => ? => ?
=> ? = 5 - 1
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [8,7,5,4,2,3,6,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,2,3,6,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,2,3,6,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [7,8,3,4,2,5,6,1] => ? => ?
=> ? = 5 - 1
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [8,6,4,3,5,2,7,1] => ? => ?
=> ? = 6 - 1
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [8,6,5,3,2,4,7,1] => ? => ?
=> ? = 6 - 1
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [8,6,4,2,3,5,7,1] => ? => ?
=> ? = 6 - 1
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [8,5,4,3,2,6,7,1] => ? => ?
=> ? = 6 - 1
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [8,5,4,2,3,6,7,1] => ? => ?
=> ? = 6 - 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [8,4,2,3,5,6,7,1] => ? => ?
=> ? = 6 - 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ?
=> ? = 6 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [6,4,3,5,7,2,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [7,5,4,6,2,3,8,1] => ? => ?
=> ? = 4 - 1
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,2,3,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [6,5,7,3,2,4,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => ? => ?
=> ? = 4 - 1
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [6,7,3,4,2,5,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [7,4,3,5,2,6,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> [7,5,4,2,3,6,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [7,5,3,2,4,6,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [7,5,2,3,4,6,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [7,3,2,4,5,6,8,1] => ? => ?
=> ? = 5 - 1
[.,[[[[[.,.],[.,.]],[.,.]],.],.]]
=> [6,4,2,3,5,7,8,1] => ? => ?
=> ? = 6 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 6 - 1
[.,[[[[[.,.],[.,[.,.]]],.],.],.]]
=> [5,4,2,3,6,7,8,1] => ? => ?
=> ? = 6 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 6 - 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6 - 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,3,1,2] => ? => ?
=> ? = 2 - 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,3,1,2] => ? => ?
=> ? = 3 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000019
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => ? => ? => ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => ? => ? => ? = 3 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [7,6,4,5,8,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ? => ? => ? = 2 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 4 - 1
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [8,7,6,3,4,5,2,1] => ? => ? => ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [8,7,3,4,5,6,2,1] => ? => ? => ? = 4 - 1
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [8,6,4,3,5,7,2,1] => ? => ? => ? = 5 - 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => ? => ? => ? = 2 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 4 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 5 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => ? = 4 - 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => [1,2,4,3,6,5,7] => ? = 3 - 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [7,6,5,4,8,2,3,1] => ? => ? => ? = 2 - 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 5 - 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 5 - 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => ? => ? => ? = 4 - 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [8,7,6,3,4,2,5,1] => ? => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => ? => ? => ? = 4 - 1
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [8,7,6,3,2,4,5,1] => ? => ? => ? = 4 - 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => ? => ? => ? = 4 - 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => ? => ? => ? = 5 - 1
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [8,7,5,4,2,3,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,2,3,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,2,3,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [7,8,3,4,2,5,6,1] => ? => ? => ? = 5 - 1
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [8,6,4,3,5,2,7,1] => ? => ? => ? = 6 - 1
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [8,6,5,3,2,4,7,1] => ? => ? => ? = 6 - 1
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [8,6,4,2,3,5,7,1] => ? => ? => ? = 6 - 1
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [8,5,4,3,2,6,7,1] => ? => ? => ? = 6 - 1
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [8,5,4,2,3,6,7,1] => ? => ? => ? = 6 - 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [8,4,2,3,5,6,7,1] => ? => ? => ? = 6 - 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ? => ? = 6 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 5 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [6,4,3,5,7,2,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [7,5,4,6,2,3,8,1] => ? => ? => ? = 4 - 1
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,2,3,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [6,5,7,3,2,4,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => ? => ? => ? = 4 - 1
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [6,7,3,4,2,5,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [7,4,3,5,2,6,8,1] => ? => ? => ? = 5 - 1
[.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> [7,5,4,2,3,6,8,1] => ? => ? => ? = 5 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [7,5,3,2,4,6,8,1] => ? => ? => ? = 5 - 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [7,5,2,3,4,6,8,1] => ? => ? => ? = 5 - 1
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St001280
Mp00016: Binary trees —left-right symmetry⟶ 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
St001280: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 90%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 90%
Values
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> []
=> ? = 1 - 1
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> []
=> ? = 1 - 1
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 1
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1 = 2 - 1
[[.,[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[[.,[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2 = 3 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 1 - 1
[[[[.,.],.],.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 5 - 1
[[[[.,.],.],[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ? = 5 - 1
[[[[.,.],[.,.]],.],[[.,.],.]]
=> [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 5 - 1
[[[[.,[.,.]],.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> ? = 5 - 1
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 5 - 1
[[[[[.,.],.],.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 5 - 1
[[[[.,.],.],[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 6 - 1
[[[[.,.],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ? = 6 - 1
[[[[.,[.,.]],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ? = 6 - 1
[[[[[.,.],.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 6 - 1
[[[[.,.],[[.,.],.]],.],[.,.]]
=> [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 6 - 1
[[[[.,[.,.]],[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 6 - 1
[[[[[.,.],.],[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 6 - 1
[[[[.,[[.,.],.]],.],.],[.,.]]
=> [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 6 - 1
[[[[[.,.],[.,.]],.],.],[.,.]]
=> [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 6 - 1
[[[[[.,[.,.]],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 6 - 1
[[[[[[.,.],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 6 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 5 - 1
[[[[.,.],[.,.]],[[.,.],.]],.]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 5 - 1
[[[[.,[.,.]],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 4 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 5 - 1
[[[[.,.],[[.,.],.]],[.,.]],.]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 5 - 1
[[[[.,[.,.]],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 5 - 1
[[[[[.,.],.],[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 5 - 1
[[[[.,[[.,.],.]],.],[.,.]],.]
=> [.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 5 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 5 - 1
[[[[[.,[.,.]],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 5 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 5 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 6 - 1
[[[[.,[.,.]],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 6 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 6 - 1
[[[[.,[[.,.],.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 6 - 1
[[[[[.,.],[.,.]],[.,.]],.],.]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 6 - 1
[[[[[.,[.,.]],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 6 - 1
[[[[.,[[[.,.],.],.]],.],.],.]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 6 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 6 - 1
[[[[[.,[.,.]],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 6 - 1
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 6 - 1
[[[[[.,[[.,.],.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 6 - 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000141
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 90%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 90%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [2,1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [2,3,1] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,4,3,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,4,1,2] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [5,8,7,6,4,3,2,1] => [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [6,5,8,7,4,3,2,1] => [1,2,3,4,7,8,5,6] => [1,2,3,4,7,5,6] => ? = 3 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => ? => ? = 3 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,5] => ? = 3 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => ? = 2 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 3 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [1,2,3,8,5,6,7,4] => [1,2,3,5,6,7,4] => ? = 2 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [1,2,3,8,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => [1,2,4,5,8,6,7,3] => [1,2,4,5,6,7,3] => ? = 2 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [1,2,4,6,7,8,5,3] => [1,2,4,6,7,5,3] => ? = 3 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [3,5,6,7,8,4,2,1] => [1,2,4,8,7,6,5,3] => [1,2,4,7,6,5,3] => ? = 4 - 1
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [3,4,8,7,6,5,2,1] => [1,2,5,6,7,8,4,3] => [1,2,5,6,7,4,3] => ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [3,4,5,8,7,6,2,1] => [1,2,6,7,8,5,4,3] => [1,2,6,7,5,4,3] => ? = 4 - 1
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4] => ? => ? = 5 - 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [1,2,8,3,4,5,6,7] => ? => ? = 1 - 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [5,7,6,4,3,8,2,1] => [1,2,8,3,4,6,7,5] => [1,2,3,4,6,7,5] => ? = 2 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [1,2,8,3,7,6,4,5] => [1,2,3,7,6,4,5] => ? = 4 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [1,2,8,7,6,5,3,4] => ? => ? = 5 - 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [1,3,4,5,7,6,2] => ? = 3 - 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [2,5,6,7,8,4,3,1] => [1,3,4,8,7,6,5,2] => [1,3,4,7,6,5,2] => ? = 4 - 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [1,3,5,6,7,8,4,2] => [1,3,5,6,7,4,2] => ? = 3 - 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => [1,3,5,7,8,6,4,2] => [1,3,5,7,6,4,2] => ? = 4 - 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [2,7,6,5,4,8,3,1] => [1,3,8,4,5,6,7,2] => ? => ? = 2 - 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [3,4,2,8,7,6,5,1] => [1,5,6,7,8,2,4,3] => [1,5,6,7,2,4,3] => ? = 4 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [2,4,3,8,7,6,5,1] => [1,5,6,7,8,3,4,2] => [1,5,6,7,3,4,2] => ? = 4 - 1
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [3,2,4,8,7,6,5,1] => [1,5,6,7,8,4,2,3] => [1,5,6,7,4,2,3] => ? = 4 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [2,5,4,3,8,7,6,1] => [1,6,7,8,3,4,5,2] => [1,6,7,3,4,5,2] => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [2,5,4,3,7,8,6,1] => [1,6,8,7,3,4,5,2] => [1,6,7,3,4,5,2] => ? = 5 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [2,4,5,3,7,8,6,1] => [1,6,8,7,3,5,4,2] => [1,6,7,3,5,4,2] => ? = 5 - 1
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [3,4,2,5,7,8,6,1] => [1,6,8,7,5,2,4,3] => [1,6,7,5,2,4,3] => ? = 5 - 1
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [4,3,6,5,2,8,7,1] => [1,7,8,2,5,6,3,4] => [1,7,2,5,6,3,4] => ? = 6 - 1
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [3,2,6,5,4,8,7,1] => [1,7,8,4,5,6,2,3] => ? => ? = 6 - 1
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,4,3,6,5,8,7,1] => [1,7,8,5,6,3,4,2] => ? => ? = 6 - 1
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [5,4,3,2,6,8,7,1] => [1,7,8,6,2,3,4,5] => [1,7,6,2,3,4,5] => ? = 6 - 1
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [2,5,4,3,6,8,7,1] => [1,7,8,6,3,4,5,2] => [1,7,6,3,4,5,2] => ? = 6 - 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [2,4,3,5,6,8,7,1] => [1,7,8,6,5,3,4,2] => [1,7,6,5,3,4,2] => ? = 6 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 3 - 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 5 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => ? = 5 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [1,8,2,7,5,6,3,4] => [1,2,7,5,6,3,4] => ? = 5 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [2,5,4,7,6,3,8,1] => [1,8,3,6,7,4,5,2] => [1,3,6,7,4,5,2] => ? = 4 - 1
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> [2,5,4,6,7,3,8,1] => [1,8,3,7,6,4,5,2] => ? => ? = 5 - 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [1,8,4,7,5,6,2,3] => [1,4,7,5,6,2,3] => ? = 5 - 1
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [3,4,2,6,7,5,8,1] => [1,8,5,7,6,2,4,3] => [1,5,7,6,2,4,3] => ? = 5 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [1,8,6,7,2,5,3,4] => [1,6,7,2,5,3,4] => ? = 5 - 1
[.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> [2,5,4,3,7,6,8,1] => [1,8,6,7,3,4,5,2] => [1,6,7,3,4,5,2] => ? = 5 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [1,8,6,7,4,5,2,3] => [1,6,7,4,5,2,3] => ? = 5 - 1
[.,[[[[[.,.],.],[.,.]],[.,.]],.]]
=> [2,3,5,4,7,6,8,1] => [1,8,6,7,4,5,3,2] => [1,6,7,4,5,3,2] => ? = 5 - 1
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [3,2,4,5,7,6,8,1] => [1,8,6,7,5,4,2,3] => [1,6,7,5,4,2,3] => ? = 5 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000740
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 70%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 1
[[.,.],.]
=> [1,2] => [1] => [1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [2,3,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [2,3,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [2,3,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [2,3,1] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [4,1,3,2] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,3,2] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [3,4,1,2] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [3,4,1,2] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [2,4,1,3] => 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,3,2] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [3,4,1,2] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,4,1,3] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [2,3,1,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [4,5,3,7,1,6,2] => ? = 2
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [5,8,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => [4,7,1,5,3,6,2] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [6,5,8,7,4,3,2,1] => [6,5,7,4,3,2,1] => [4,7,1,6,2,5,3] => ? = 3
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [4,7,1,6,2,5,3] => ? = 3
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [4,6,2,7,1,5,3] => ? = 3
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => [7,1,4,5,3,6,2] => ? = 2
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [4,7,1,6,2,5,3] => ? = 3
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [4,7,6,5,3,2,1] => [7,1,4,5,3,6,2] => ? = 2
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [6,2,5,3,7,1,4] => ? = 4
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => [3,7,6,5,4,2,1] => [5,4,7,1,3,6,2] => ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => [6,2,5,4,7,1,3] => ? = 3
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [3,5,6,7,8,4,2,1] => [3,5,6,7,4,2,1] => [7,1,3,6,2,5,4] => ? = 4
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [3,4,8,7,6,5,2,1] => [3,4,7,6,5,2,1] => [5,6,2,4,7,1,3] => ? = 3
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [3,4,5,8,7,6,2,1] => [3,4,5,7,6,2,1] => [7,1,3,5,6,2,4] => ? = 4
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [4,3,6,5,8,7,2,1] => [4,3,6,5,7,2,1] => [6,2,3,7,1,4,5] => ? = 5
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [5,7,6,4,3,8,2,1] => [5,7,6,4,3,2,1] => [4,7,1,5,3,6,2] => ? = 2
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [7,1,5,3,6,2,4] => ? = 4
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [7,1,4,6,2,3,5] => ? = 5
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => [5,4,6,3,7,1,2] => ? = 2
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [2,6,7,5,4,3,1] => [5,4,7,1,2,6,3] => ? = 3
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [2,5,6,7,8,4,3,1] => [2,5,6,7,4,3,1] => [6,3,7,1,2,5,4] => ? = 4
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [2,4,7,6,5,3,1] => [5,7,1,2,4,6,3] => ? = 3
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => [2,4,6,7,5,3,1] => [5,6,3,7,1,2,4] => ? = 4
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [2,7,6,5,4,8,3,1] => [2,7,6,5,4,3,1] => [5,4,6,3,7,1,2] => ? = 2
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [2,3,5,6,8,7,4,1] => [2,3,5,6,7,4,1] => [6,4,7,1,2,3,5] => ? = 5
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [2,3,5,6,7,8,4,1] => [2,3,5,6,7,4,1] => [6,4,7,1,2,3,5] => ? = 5
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [4,3,2,8,7,6,5,1] => [4,3,2,7,6,5,1] => [3,2,6,5,7,1,4] => ? = 4
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [3,4,2,8,7,6,5,1] => [3,4,2,7,6,5,1] => [6,5,7,1,3,2,4] => ? = 4
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [2,4,3,8,7,6,5,1] => [2,4,3,7,6,5,1] => [3,6,5,7,1,2,4] => ? = 4
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [3,2,4,8,7,6,5,1] => [3,2,4,7,6,5,1] => [2,6,5,7,1,3,4] => ? = 4
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [2,3,4,8,7,6,5,1] => [2,3,4,7,6,5,1] => [6,5,7,1,2,3,4] => ? = 4
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => [5,4,3,2,7,6,1] => [3,4,2,6,7,1,5] => ? = 5
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [5,4,3,2,7,8,6,1] => [5,4,3,2,7,6,1] => [3,4,2,6,7,1,5] => ? = 5
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [2,5,4,3,8,7,6,1] => [2,5,4,3,7,6,1] => [4,3,6,7,1,2,5] => ? = 5
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [2,5,4,3,7,8,6,1] => [2,5,4,3,7,6,1] => [4,3,6,7,1,2,5] => ? = 5
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [2,4,5,3,7,8,6,1] => [2,4,5,3,7,6,1] => [6,7,1,2,4,3,5] => ? = 5
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [3,4,2,5,7,8,6,1] => [3,4,2,5,7,6,1] => [6,7,1,3,2,4,5] => ? = 5
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [4,3,6,5,2,8,7,1] => [4,3,6,5,2,7,1] => [7,1,4,5,2,3,6] => ? = 6
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [3,2,6,5,4,8,7,1] => [3,2,6,5,4,7,1] => [2,5,4,7,1,3,6] => ? = 6
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,4,3,6,5,8,7,1] => [2,4,3,6,5,7,1] => [3,5,7,1,2,4,6] => ? = 6
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [5,4,3,2,6,8,7,1] => [5,4,3,2,6,7,1] => [3,4,2,7,1,5,6] => ? = 6
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [2,5,4,3,6,8,7,1] => [2,5,4,3,6,7,1] => [4,3,7,1,2,5,6] => ? = 6
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [2,4,3,5,6,8,7,1] => [2,4,3,5,6,7,1] => [3,7,1,2,4,5,6] => ? = 6
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,8,7,1] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 6
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000957
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 65%●distinct values known / distinct values provided: 60%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 65%●distinct values known / distinct values provided: 60%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 2 - 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => ? => ? => ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => ? => ? => ? = 3 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 3 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 3 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 3 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 4 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ? => ? => ? = 2 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 4 - 1
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [8,7,6,3,4,5,2,1] => ? => ? => ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [8,7,3,4,5,6,2,1] => ? => ? => ? = 4 - 1
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [8,6,4,3,5,7,2,1] => ? => ? => ? = 5 - 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => ? => ? => ? = 2 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 4 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 5 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => ? = 4 - 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [7,6,5,4,8,2,3,1] => ? => ? => ? = 2 - 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 5 - 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 5 - 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => ? => ? => ? = 4 - 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [8,7,6,3,4,2,5,1] => ? => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => ? => ? => ? = 4 - 1
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [8,7,6,3,2,4,5,1] => ? => ? => ? = 4 - 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => ? => ? => ? = 4 - 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => ? => ? => ? = 5 - 1
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [8,7,5,4,2,3,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,2,3,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,2,3,6,1] => ? => ? => ? = 5 - 1
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [7,8,3,4,2,5,6,1] => ? => ? => ? = 5 - 1
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [8,6,4,3,5,2,7,1] => ? => ? => ? = 6 - 1
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [8,6,5,3,2,4,7,1] => ? => ? => ? = 6 - 1
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [8,6,4,2,3,5,7,1] => ? => ? => ? = 6 - 1
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [8,5,4,3,2,6,7,1] => ? => ? => ? = 6 - 1
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [8,5,4,2,3,6,7,1] => ? => ? => ? = 6 - 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [8,4,2,3,5,6,7,1] => ? => ? => ? = 6 - 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ? => ? = 6 - 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 2 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 3 - 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [2,3,4,5,1,6,7] => ? = 5 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 5 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [6,4,3,5,7,2,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [7,5,4,6,2,3,8,1] => ? => ? => ? = 4 - 1
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,2,3,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [6,5,7,3,2,4,8,1] => ? => ? => ? = 5 - 1
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => ? => ? => ? = 4 - 1
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000653
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 70%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 70%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,2] => [1] => ? = 1 - 1
[[.,.],.]
=> [1,0,1,0]
=> [2,1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 1 = 2 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => ? = 2 - 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 3 - 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? = 1 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 3 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 3 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7] => ? = 2 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 3 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [5,1,2,8,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 4 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,1,8,2,3,5,6,7] => ? => ? = 2 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,6,1,2,3,5,7,8] => [4,6,1,2,3,5,7] => ? = 3 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,6,7,8,1,2,3,5] => [4,6,7,1,2,3,5] => ? = 4 - 1
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,5,1,2,3,6,7,8] => ? => ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,1,2,3,7,8] => [4,5,6,1,2,3,7] => ? = 4 - 1
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,7,3,4,6,8] => [1,5,2,7,3,4,6] => ? = 5 - 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,8,4,5,7] => ? => ? = 2 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,6,7,2,8,3,4,5] => [1,6,7,2,3,4,5] => ? = 4 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,2,3,4] => [1,5,6,7,2,3,4] => ? = 5 - 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6,8] => [3,7,1,2,4,5,6] => ? = 3 - 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,6,7,8,1,2,4,5] => [3,6,7,1,2,4,5] => ? = 4 - 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,5,1,2,4,6,7,8] => ? => ? = 3 - 1
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6,8] => ? => ? = 4 - 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [3,1,2,4,8,5,6,7] => [3,1,2,4,5,6,7] => ? = 2 - 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,7,1,2,5,8] => [3,4,6,7,1,2,5] => ? = 5 - 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,6,7,8,1,2,5] => ? => ? = 5 - 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,5,3,4,6,7,8] => ? => ? = 4 - 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [4,1,5,2,3,6,7,8] => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,5,2,4,6,7,8] => ? => ? = 4 - 1
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,4,5,2,3,6,7,8] => ? => ? = 4 - 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7,8] => ? => ? = 4 - 1
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,2,3,6,8,4,5,7] => ? => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,1,2,6,4,5,7,8] => ? => ? = 5 - 1
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [3,1,2,6,8,4,5,7] => ? => ? = 5 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [3,5,1,6,8,2,4,7] => ? => ? = 5 - 1
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [4,1,5,6,8,2,3,7] => [4,1,5,6,2,3,7] => ? = 5 - 1
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [1,5,2,3,7,4,6,8] => [1,5,2,3,7,4,6] => ? = 6 - 1
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,7,4,6,8] => [3,1,5,2,7,4,6] => ? = 6 - 1
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,2,3,6,7,4,5,8] => ? => ? = 6 - 1
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [3,1,2,6,7,4,5,8] => ? => ? = 6 - 1
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,1,5,6,7,2,4,8] => ? => ? = 6 - 1
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,7,1,2,8] => ? => ? = 6 - 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? = 3 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,2,7,3,8,4,5] => [1,6,2,7,3,4,5] => ? = 5 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,7,3,8,4,6] => [1,5,2,7,3,4,6] => ? = 5 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,1,6,2,4,8,5,7] => ? => ? = 4 - 1
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000727
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 70%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 1
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [3,2,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [1,3,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [1,3,2] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [3,1,2] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [3,1,2] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [1,3,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [3,1,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [1,4,3,2] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [1,4,3,2] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [2,1,4,3] => 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [1,2,4,3] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [1,4,3,2] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,1,4,3] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [1,2,4,3] => 3
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [1,4,2,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [1,4,2,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [1,3,2,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [1,4,3,2] => 2
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1] => [4,1,3,2] => 2
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 2
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [5,8,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 2
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [6,5,8,7,4,3,2,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 3
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 3
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 2
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 3
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [4,7,6,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 2
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => [3,7,6,5,4,2,1] => [7,6,5,1,4,3,2] => ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => [7,2,6,1,5,4,3] => ? = 3
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [3,5,6,7,8,4,2,1] => [3,5,6,7,4,2,1] => [2,3,7,1,6,5,4] => ? = 4
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [3,4,8,7,6,5,2,1] => [3,4,7,6,5,2,1] => [7,6,1,2,5,4,3] => ? = 3
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [3,4,5,8,7,6,2,1] => [3,4,5,7,6,2,1] => [7,1,2,3,6,5,4] => ? = 4
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [5,7,6,4,3,8,2,1] => [5,7,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 2
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 4
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ? = 5
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => [7,6,5,4,1,3,2] => ? = 2
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [2,6,7,5,4,3,1] => [2,7,6,5,1,4,3] => ? = 3
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [2,5,6,7,8,4,3,1] => [2,5,6,7,4,3,1] => [2,3,7,6,1,5,4] => ? = 4
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [2,4,7,6,5,3,1] => [7,6,2,5,1,4,3] => ? = 3
[.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => [2,4,6,7,5,3,1] => [3,7,2,6,1,5,4] => ? = 4
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [2,7,6,5,4,8,3,1] => [2,7,6,5,4,3,1] => [7,6,5,4,1,3,2] => ? = 2
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [2,3,5,6,8,7,4,1] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ? = 5
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [2,3,5,6,7,8,4,1] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ? = 5
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [4,3,2,8,7,6,5,1] => [4,3,2,7,6,5,1] => [1,7,2,6,3,5,4] => ? = 4
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [3,4,2,8,7,6,5,1] => [3,4,2,7,6,5,1] => [7,1,2,6,3,5,4] => ? = 4
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [2,4,3,8,7,6,5,1] => [2,4,3,7,6,5,1] => [7,2,6,1,3,5,4] => ? = 4
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [3,2,4,8,7,6,5,1] => [3,2,4,7,6,5,1] => [7,1,6,2,3,5,4] => ? = 4
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [2,3,4,8,7,6,5,1] => [2,3,4,7,6,5,1] => [7,6,1,2,3,5,4] => ? = 4
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 5
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [5,4,3,2,7,8,6,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 5
[.,[[[.,.],[.,[.,.]]],[.,[.,.]]]]
=> [2,5,4,3,8,7,6,1] => [2,5,4,3,7,6,1] => [3,2,7,1,4,6,5] => ? = 5
[.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [2,5,4,3,7,8,6,1] => [2,5,4,3,7,6,1] => [3,2,7,1,4,6,5] => ? = 5
[.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [2,4,5,3,7,8,6,1] => [2,4,5,3,7,6,1] => [2,3,7,1,4,6,5] => ? = 5
[.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [4,3,6,5,2,8,7,1] => [4,3,6,5,2,7,1] => [1,5,2,4,3,7,6] => ? = 6
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [3,2,6,5,4,8,7,1] => [3,2,6,5,4,7,1] => [5,1,4,2,3,7,6] => ? = 6
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,4,3,6,5,8,7,1] => [2,4,3,6,5,7,1] => [2,5,1,3,4,7,6] => ? = 6
[.,[[[.,[.,[.,[.,.]]]],.],[.,.]]]
=> [5,4,3,2,6,8,7,1] => [5,4,3,2,6,7,1] => [4,3,2,1,5,7,6] => ? = 6
[.,[[[[.,.],[.,[.,.]]],.],[.,.]]]
=> [2,5,4,3,6,8,7,1] => [2,5,4,3,6,7,1] => [4,3,1,2,5,7,6] => ? = 6
[.,[[[[[.,.],[.,.]],.],.],[.,.]]]
=> [2,4,3,5,6,8,7,1] => [2,4,3,5,6,7,1] => [3,1,2,4,5,7,6] => ? = 6
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 2
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 3
Description
The largest label of a leaf in the binary search tree associated with the permutation.
Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St001557The number of inversions of the second entry of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!