Your data matches 158 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001007
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000024
(load all 82 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000676
(load all 21 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1,0]
 => 1
[[],[]]
 => [[.,.],.]
 => [1,1,0,0]
 => 1
[[[]]]
 => [.,[.,.]]
 => [1,0,1,0]
 => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 3
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => 3
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 3
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001068
(load all 81 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
(load all 81 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000211
(load all 20 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => {{1}}
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => {{1},{2}}
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => {{1,2}}
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 3 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2 = 3 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2 = 3 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2 = 3 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2 = 3 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3 = 4 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2 = 3 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3 = 4 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 3 = 4 - 1
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => [1]
 => 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => [2]
 => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => [1,1]
 => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3]
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [2,1]
 => 2
[[[]],[]]
 => [[.,.],[.,.]]
 => [1,3,2] => [2,1]
 => 2
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1]
 => 2
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,1,1]
 => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4]
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,1]
 => 2
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,1]
 => 2
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,1]
 => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,1,1]
 => 3
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [3,1]
 => 2
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,1,1]
 => 3
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => 2
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,1,1]
 => 3
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1]
 => 2
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,1,1]
 => 3
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [2,1,1]
 => 3
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,1]
 => 3
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5]
 => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,1]
 => 2
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [4,1]
 => 2
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,1]
 => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,1,1]
 => 3
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [4,1]
 => 2
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,1,1]
 => 3
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2]
 => 2
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,1,1]
 => 3
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,1]
 => 2
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,1,1]
 => 3
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [3,1,1]
 => 3
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,1,1]
 => 3
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,1,1,1]
 => 4
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [4,1]
 => 2
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [3,1,1]
 => 3
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [3,1,1]
 => 3
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [3,1,1]
 => 3
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,2]
 => 2
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [3,1,1]
 => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [2,1,1,1]
 => 4
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2]
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,2,1]
 => 3
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,2,1]
 => 3
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,1,1,1]
 => 4
Description
The length of the partition.
Matching statistic: St000093
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [1,1,0,0]
 => [2] => ([],2)
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 4
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000105
(load all 9 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => {{1}}
 => 1
[[],[]]
 => [1,0,1,0]
 => [2,1] => {{1,2}}
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,2] => {{1},{2}}
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,3,1] => {{1,2,3}}
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,1,3] => {{1,2},{3}}
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,3,2] => {{1},{2,3}}
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [3,1,2] => {{1,3},{2}}
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => {{1,2,4},{3},{5}}
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 3
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => {{1},{2,3,5},{4}}
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 4
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => {{1,3,4},{2},{5}}
 => 3
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 4
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => {{1,3},{2,4,5}}
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => {{1,3},{2},{4,5}}
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => {{1},{2,4,5},{3}}
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 4
Description
The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000147
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => [1]
 => 1
[[],[]]
 => [[.,.],.]
 => [1,2] => [1,1]
 => 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [2]
 => 2
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [1,1,1]
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [2,1]
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1]
 => 2
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [2,1]
 => 2
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3]
 => 3
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,1,1,1]
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [2,1,1]
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [2,1,1]
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,1]
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1]
 => 3
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,1]
 => 2
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1]
 => 3
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,1,1]
 => 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1]
 => 3
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,1,1]
 => 2
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [3,1]
 => 3
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,1]
 => 3
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,1]
 => 3
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4]
 => 4
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [2,1,1,1]
 => 2
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,1,1]
 => 3
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [2,1,1,1]
 => 2
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,1]
 => 3
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [2,1,1,1]
 => 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [3,1,1]
 => 3
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,1,1]
 => 3
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,1,1]
 => 3
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [3,1,1]
 => 3
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,1]
 => 4
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [3,1,1]
 => 3
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [3,1,1]
 => 3
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,1]
 => 3
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,1]
 => 4
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [2,1,1,1]
 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,1,1]
 => 3
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [3,1,1]
 => 3
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1]
 => 4
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [3,1,1]
 => 3
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,1,1]
 => 3
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,1,1]
 => 3
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1]
 => 4
Description
The largest part of an integer partition.
The following 148 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000507The number of ascents of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001581The achromatic number of a graph. St000157The number of descents of a standard tableau. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. St000288The number of ones in a binary word. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000362The size of a minimal vertex cover of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001304The number of maximally independent sets of vertices of a graph. St000527The width of the poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000632The jump number of the poset. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000167The number of leaves of an ordered tree. St000996The number of exclusive left-to-right maxima of a permutation. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000245The number of ascents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000702The number of weak deficiencies of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000308The height of the tree associated to a permutation. St000662The staircase size of the code of a permutation. St001622The number of join-irreducible elements of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000052The number of valleys of a Dyck path not on the x-axis. St001875The number of simple modules with projective dimension at most 1. St000031The number of cycles in the cycle decomposition of a permutation. St000141The maximum drop size of a permutation. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000332The positive inversions of an alternating sign matrix. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000069The number of maximal elements of a poset. St001427The number of descents of a signed permutation. St000542The number of left-to-right-minima of a permutation. St000377The dinv defect of an integer partition. St000168The number of internal nodes of an ordered tree. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000080The rank of the poset. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001820The size of the image of the pop stack sorting operator. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000155The number of exceedances (also excedences) of a permutation. St000386The number of factors DDU in a Dyck path. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000061The number of nodes on the left branch of a binary tree. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000619The number of cyclic descents of a permutation. St000242The number of indices that are not cyclical small weak excedances. St001152The number of pairs with even minimum in a perfect matching. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001896The number of right descents of a signed permutations. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001720The minimal length of a chain of small intervals in a lattice.