Your data matches 73 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000251
(load all 12 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => {{1,2}}
 => 1
[[.,.],.]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000142
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
Description
The number of even parts of a partition.
Matching statistic: St000157
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => [[1],[2]]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => [[1,2]]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [[1,3],[2]]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [[1,2,4],[3]]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [[1,3,5],[2,4]]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [[1,3,4],[2,5]]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [[1,2,4,5],[3]]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [[1,2,3],[4,5]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [[1,2,4],[3,5]]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [[1,2,3,5],[4]]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,4],[5]]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [[1,3,4,5],[2]]
 => 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => 10 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => 01 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 1000 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => 0100 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 1010 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => 1010 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 1001 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 1001 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 1000 => 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000389
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => 10 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => 01 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 1000 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => 0100 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 1010 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => 1010 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 1001 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 1001 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 1000 => 1
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
(load all 4 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => 10 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => 01 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 1000 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => 0100 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 1010 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => 1010 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 1001 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 1001 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 1000 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000566
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St001251
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
Description
The number of parts of an integer partition that are at least two.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001657The number of twos in an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000919The number of maximal left branches of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000211The rank of the set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St001737The number of descents of type 2 in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000386The number of factors DDU in a Dyck path. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000703The number of deficiencies of a permutation. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000035The number of left outer peaks of a permutation. St001427The number of descents of a signed permutation. St000702The number of weak deficiencies of a permutation. St000245The number of ascents of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000325The width of the tree associated to a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001840The number of descents of a set partition. St001874Lusztig's a-function for the symmetric group. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St000647The number of big descents of a permutation. St000646The number of big ascents of a permutation. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St000092The number of outer peaks of a permutation. St001864The number of excedances of a signed permutation. St000534The number of 2-rises of a permutation. St001597The Frobenius rank of a skew partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000920The logarithmic height of a Dyck path.