Your data matches 46 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000254
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => {{1,2}}
 => 1
[[.,.],.]
 => [1,2] => {{1},{2}}
 => 0
[.,[.,[.,.]]]
 => [3,2,1] => {{1,3},{2}}
 => 1
[.,[[.,.],.]]
 => [2,3,1] => {{1,2,3}}
 => 1
[[.,.],[.,.]]
 => [3,1,2] => {{1,3},{2}}
 => 1
[[.,[.,.]],.]
 => [2,1,3] => {{1,2},{3}}
 => 1
[[[.,.],.],.]
 => [1,2,3] => {{1},{2},{3}}
 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => {{1,4},{2,3}}
 => 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => {{1,3},{2,4}}
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => {{1,4},{2},{3}}
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => {{1,3,4},{2}}
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => {{1,2,3,4}}
 => 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => {{1,4},{2,3}}
 => 2
[[.,.],[[.,.],.]]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => {{1,4},{2},{3}}
 => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => {{1,3},{2},{4}}
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => {{1,4},{2,5},{3}}
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => {{1,5},{2,3,4}}
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => {{1,4},{2,3,5}}
 => 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => {{1,3,5},{2,4}}
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => {{1,5},{2,4},{3}}
 => 2
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => {{1,4},{2,5},{3}}
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 2
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => {{1,4,5},{2,3}}
 => 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => {{1,3},{2,4,5}}
 => 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => {{1,4,5},{2},{3}}
 => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => {{1,3,4,5},{2}}
 => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => {{1,5},{2,4},{3}}
 => 2
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => {{1,4},{2,5},{3}}
 => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => 2
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => {{1,4},{2,3,5}}
 => 2
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => {{1,3,5},{2,4}}
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => {{1,5},{2,4},{3}}
 => 2
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => {{1,4},{2,5},{3}}
 => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => {{1,5},{2,4},{3}}
 => 2
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => {{1,4},{2,5},{3}}
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => {{1,5},{2,3},{4}}
 => 2
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
 => 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => 2
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
 => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => {{1,4},{2,3},{5}}
 => 2
Description
The nesting number of a set partition. This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000142
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
Description
The number of even parts of a partition.
Matching statistic: St000157
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [[1],[2]]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [[1,2]]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [[1,3],[2]]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [[1,2,4],[3]]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [[1,2,5],[3,4]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [[1,3,5],[2,4]]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [[1,3,5],[2,4]]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [[1,3,4],[2,5]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [[1,3,4,5],[2]]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 2
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
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => 10 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => 01 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => 10 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => 01 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 101 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 100 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 010 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 101 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 010 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 010 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 001 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 001 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 1010 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 0100 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 0010 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 1010 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 0100 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 1001 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 0101 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 0100 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 1010 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 0100 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 1000 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 1001 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 1000 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000389
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => 10 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => 01 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => 10 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => 01 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 101 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 100 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 010 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 101 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 010 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 010 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 001 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 001 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 1010 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 0100 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 0010 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 1010 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 0100 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 1001 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 0101 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 0100 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 1010 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 0100 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 1000 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 1001 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 1000 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => 10 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => 01 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => 10 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => 01 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 101 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 010 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 100 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 010 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 101 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 010 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 010 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 001 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 001 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 1010 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 0100 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 0100 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 0010 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 1010 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 0100 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 1001 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 0101 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 0100 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 1010 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 0100 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 1000 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => 1001 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 1000 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000566
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
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
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
Mapping
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 2
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
Description
The number of parts of an integer partition that are at least two.
The following 36 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. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000919The number of maximal left branches of a binary tree. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000884The number of isolated descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000662The staircase size of the code of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000035The number of left outer peaks of a permutation. St001665The number of pure excedances of a permutation. St001427The number of descents of a signed permutation. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000619The number of cyclic descents of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000920The logarithmic height of a Dyck path.