Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001092
(load all 3 compositions to match this statistic)
Mapping
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 0
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 1
[5,1]
 => 0
[4,2]
 => 2
[4,1,1]
 => 1
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 0
[4,3]
 => 1
[4,2,1]
 => 2
[4,1,1,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 1
[6,2]
 => 2
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 1
[5,1,1,1]
 => 0
[4,4]
 => 1
[4,3,1]
 => 1
[4,2,1,1]
 => 2
[4,1,1,1,1]
 => 1
[3,2,1,1,1]
 => 1
[3,1,1,1,1,1]
 => 0
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St000257
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0
[2]
 => [1,1]
 => 1
[1,1]
 => [2]
 => 0
[3]
 => [3]
 => 0
[2,1]
 => [1,1,1]
 => 1
[1,1,1]
 => [2,1]
 => 0
[4]
 => [2,2]
 => 1
[3,1]
 => [3,1]
 => 0
[2,2]
 => [1,1,1,1]
 => 1
[2,1,1]
 => [2,1,1]
 => 1
[1,1,1,1]
 => [4]
 => 0
[5]
 => [5]
 => 0
[4,1]
 => [2,2,1]
 => 1
[3,2]
 => [3,1,1]
 => 1
[3,1,1]
 => [3,2]
 => 0
[2,2,1]
 => [1,1,1,1,1]
 => 1
[2,1,1,1]
 => [2,1,1,1]
 => 1
[1,1,1,1,1]
 => [4,1]
 => 0
[6]
 => [3,3]
 => 1
[5,1]
 => [5,1]
 => 0
[4,2]
 => [2,2,1,1]
 => 2
[4,1,1]
 => [2,2,2]
 => 1
[3,2,1]
 => [3,1,1,1]
 => 1
[3,1,1,1]
 => [3,2,1]
 => 0
[2,2,1,1]
 => [2,1,1,1,1]
 => 1
[2,1,1,1,1]
 => [4,1,1]
 => 1
[1,1,1,1,1,1]
 => [4,2]
 => 0
[6,1]
 => [3,3,1]
 => 1
[5,2]
 => [5,1,1]
 => 1
[5,1,1]
 => [5,2]
 => 0
[4,3]
 => [3,2,2]
 => 1
[4,2,1]
 => [2,2,1,1,1]
 => 2
[4,1,1,1]
 => [2,2,2,1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => 1
[3,2,1,1]
 => [3,2,1,1]
 => 1
[3,1,1,1,1]
 => [4,3]
 => 0
[2,1,1,1,1,1]
 => [4,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => 0
[8]
 => [4,4]
 => 1
[6,2]
 => [3,3,1,1]
 => 2
[6,1,1]
 => [3,3,2]
 => 1
[5,3]
 => [5,3]
 => 0
[5,2,1]
 => [5,1,1,1]
 => 1
[5,1,1,1]
 => [5,2,1]
 => 0
[4,4]
 => [2,2,2,2]
 => 1
[4,3,1]
 => [3,2,2,1]
 => 1
[4,2,1,1]
 => [2,2,2,1,1]
 => 2
[4,1,1,1,1]
 => [4,2,2]
 => 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => 1
[3,1,1,1,1,1]
 => [4,3,1]
 => 0
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000023
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 0
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 0
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 0
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 0
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 0
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 0
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 2
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 0
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 0
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 0
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 2
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 0
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 0
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 0
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 0
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => 2
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 0
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Matching statistic: St000196
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 0
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 0
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 0
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 0
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 0
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 0
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 0
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 0
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => 0
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 0
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 0
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => 0
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => 2
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 0
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 0
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 2
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => 0
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => 0
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => 2
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 0
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000386
(load all 2 compositions to match this statistic)
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 0
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 0
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 0
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 2
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000779
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 0
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 0
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 0
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 0
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 0
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 0
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 2
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 0
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 0
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 0
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 2
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 0
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 0
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 0
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 0
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => 2
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 0
Description
The tier of a permutation. This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$. According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St000099
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1 = 0 + 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2 = 1 + 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1 = 0 + 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1 = 0 + 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2 = 1 + 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1 = 0 + 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2 = 1 + 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1 = 0 + 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 2 = 1 + 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2 = 1 + 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 1 = 0 + 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2 = 1 + 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2 = 1 + 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1 = 0 + 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 2 = 1 + 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2 = 1 + 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1 = 0 + 1
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 2 = 1 + 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 1 = 0 + 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 3 = 2 + 1
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2 = 1 + 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 2 = 1 + 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1 = 0 + 1
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 2 = 1 + 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 2 = 1 + 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 1 = 0 + 1
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 2 = 1 + 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 2 = 1 + 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 1 = 0 + 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 2 = 1 + 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 3 = 2 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 2 = 1 + 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 2 = 1 + 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1 = 0 + 1
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1 = 0 + 1
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => 2 = 1 + 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 3 = 2 + 1
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2 = 1 + 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 1 = 0 + 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 2 = 1 + 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 1 = 0 + 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => 2 = 1 + 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 2 = 1 + 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => 3 = 2 + 1
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2 = 1 + 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => 2 = 1 + 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 1 = 0 + 1
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000201
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 1 = 0 + 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 1 = 0 + 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 0 + 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 1 = 0 + 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 2 = 1 + 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1 = 0 + 1
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => 1 = 0 + 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 3 = 2 + 1
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 1 + 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 2 = 1 + 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 1 = 0 + 1
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => 2 = 1 + 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => 1 = 0 + 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => 3 = 2 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => 2 = 1 + 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 1 = 0 + 1
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => 2 = 1 + 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 1 + 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => 1 = 0 + 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => 2 = 1 + 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => 1 = 0 + 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => 3 = 2 + 1
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 1 + 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 2 = 1 + 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 1 = 0 + 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000659
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 0 + 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 1 = 0 + 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[6]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => 1 = 0 + 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[6,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[8]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[6,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[6,1,1]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[3,1,1,1,1,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001115
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,2,4,5,1] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,2,3,1] => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,2,4,5,6,1] => 0
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,4,2,5,1] => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,2,3,5,1] => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,2,3,1,4] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [6,2,1,3,4,5] => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 0
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [4,2,3,5,6,1] => 0
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,4,5,2,1] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,2,5,1] => 2
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,2,3,4,1] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,5,2,1,3] => 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [6,2,3,1,4,5] => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => 0
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [4,2,3,5,6,7,1] => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,4,5,2,6,1] => 0
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,3,2,5,6,1] => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [5,2,3,4,6,1] => 0
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [3,4,5,6,1,2] => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,3,5,2,1] => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,2,4,1] => 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [6,2,3,4,1,5] => 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [6,3,2,1,4,5] => 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => [7,2,3,1,4,5,6] => 0
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [9,4,1,2,3,5,6,7,8] => [3,4,5,2,6,7,8,9,1] => ? = 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => [4,3,2,5,6,7,8,9,1] => ? = 2
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,4,1,5,6,7,8] => [5,2,3,4,6,7,8,9,1] => ? = 1
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,3,4,6] => [4,3,5,6,2,7,1] => ? = 2
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => [4,5,3,2,6,7,1] => ? = 2
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,3,1,5,6] => [5,3,4,2,6,7,1] => ? = 1
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [7,2,3,4,5,6,1] => ? = 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => [7,3,2,4,5,1,6] => ? = 1
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => [8,2,3,4,5,1,6,7] => ? = 0
[4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,7,1] => [7,3,4,2,1,5,6] => ? = 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => [9,2,3,4,1,5,6,7,8] => ? = 1
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => [9,3,2,1,4,5,6,7,8] => ? = 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ? => ? = 2
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ? => ? = 1
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ? => ? = 2
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1,3,4,5] => [4,3,5,6,7,2,1] => ? = 1
[6,3,1,1,1]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,3,5,1,6] => [6,3,4,2,5,7,1] => ? = 1
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => ? = 2
[6,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [7,2,3,4,5,6,8,1] => [8,2,3,4,5,6,1,7] => ? = 1
[5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [6,4,2,3,5,7,1] => [7,3,4,2,5,1,6] => ? = 0
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => [8,3,2,4,5,1,6,7] => ? = 1
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => [9,2,3,4,5,1,6,7,8] => ? = 0
[4,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,2,3,6,7,8,1] => [8,3,4,2,1,5,6,7] => ? = 1
[8,4,1]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
 => [9,5,2,1,3,4,6,7,8] => ? => ? = 2
[8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [9,4,3,1,2,5,6,7,8] => [4,5,3,2,6,7,8,9,1] => ? = 2
[8,3,1,1]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,3,1,5,6,7,8] => ? => ? = 1
[6,5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [7,6,3,1,2,4,5] => [4,5,3,6,7,2,1] => ? = 2
[6,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,1,4,5] => [5,3,4,6,7,2,1] => ? = 1
[6,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [7,2,3,4,5,6,8,9,1] => [9,2,3,4,5,6,1,7,8] => ? = 1
[5,4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [6,5,2,3,4,7,1] => [7,3,4,5,2,1,6] => ? = 1
[5,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [6,4,2,3,5,7,8,1] => [8,3,4,2,5,1,6,7] => ? = 0
[5,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [6,3,2,4,5,7,8,9,1] => ? => ? = 1
[8,3,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? => ? => ? = 1
[6,5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,1,5] => [6,3,4,5,7,2,1] => ? = 1
[5,4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [6,5,2,3,4,7,8,1] => [8,3,4,5,2,1,6,7] => ? = 1
[5,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [6,4,3,2,5,7,8,1] => [8,4,3,2,5,1,6,7] => ? = 1
[5,3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [6,4,2,3,5,7,8,9,1] => ? => ? = 0
[5,3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
 => ? => ? => ? = 0
Description
The number of even descents of a permutation.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000256The number of parts from which one can substract 2 and still get an integer partition. St000353The number of inner valleys of a permutation. St000092The number of outer peaks of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001729The number of visible descents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.