- FindStat

Your data matches 47 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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] => [1]
 => 0
[.,[.,.]]
 => [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

Description

The number of even parts of a partition.

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] => [1]
 => 0
[.,[.,.]]
 => [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

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] => [1]
 => 0
[.,[.,.]]
 => [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

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] => [1]
 => 0
[.,[.,.]]
 => [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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St001657

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
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1] => [1]
 => 0
[.,[.,.]]
 => [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

Description

The number of twos in an integer partition. The total number of twos in all partitions of

$n$ is equal to the total number of singletons [[St001484]] in all partitions of

$n-1$ , see [1].

Matching statistic: St000345

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
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1] => [1]
 => 1 = 0 + 1
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1

Description

The number of refinements of a partition. A partition

$\lambda$ refines a partition

$\mu$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

Matching statistic: St000935

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
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1] => [1]
 => 1 = 0 + 1
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1

Description

The number of ordered refinements of an integer partition. This is, for an integer partition

$\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition

$\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices

$1 = a_0 < \ldots < a_n = m$ with

$\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$ .

Matching statistic: St001389

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
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1] => [1]
 => 1 = 0 + 1
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

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] => [1]
 => ? = 0
[.,[.,.]]
 => [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: St000321

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
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1] => [1]
 => 1 = 0 + 1
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,2]
 => 3 = 2 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2 = 1 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,2]
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,2,1]
 => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4,7,6,9,8,11,10] => [2,2,2,2,2,1]
 => ? = 5 + 1
[[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.],.]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0 + 1

Description

The number of integer partitions of n that are dominated by an integer partition. A partition

$\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition

$\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if

$\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all

$k$ .

The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000157The number of descents of a standard tableau. St000919The number of maximal left branches of a binary tree. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. 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. St000035The number of left outer peaks 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. St000996The number of exclusive left-to-right maxima of a permutation. St000662The staircase size of the code of a permutation. St000245The number of ascents of a permutation. St000834The number of right outer peaks 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001427The number of descents of a signed 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. St001624The breadth of a lattice.