searching the database
Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000974
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1
[[],[]]
=> 0
[[[]]]
=> 2
[[],[],[]]
=> 0
[[],[[]]]
=> 0
[[[]],[]]
=> 0
[[[],[]]]
=> 1
[[[[]]]]
=> 3
[[],[],[],[]]
=> 0
[[],[],[[]]]
=> 0
[[],[[]],[]]
=> 0
[[],[[],[]]]
=> 0
[[],[[[]]]]
=> 0
[[[]],[],[]]
=> 0
[[[]],[[]]]
=> 0
[[[],[]],[]]
=> 0
[[[[]]],[]]
=> 0
[[[],[],[]]]
=> 1
[[[],[[]]]]
=> 1
[[[[]],[]]]
=> 1
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 4
[[],[],[],[],[]]
=> 0
[[],[],[],[[]]]
=> 0
[[],[],[[]],[]]
=> 0
[[],[],[[],[]]]
=> 0
[[],[],[[[]]]]
=> 0
[[],[[]],[],[]]
=> 0
[[],[[]],[[]]]
=> 0
[[],[[],[]],[]]
=> 0
[[],[[[]]],[]]
=> 0
[[],[[],[],[]]]
=> 0
[[],[[],[[]]]]
=> 0
[[],[[[]],[]]]
=> 0
[[],[[[],[]]]]
=> 0
[[],[[[[]]]]]
=> 0
[[[]],[],[],[]]
=> 0
[[[]],[],[[]]]
=> 0
[[[]],[[]],[]]
=> 0
[[[]],[[],[]]]
=> 0
[[[]],[[[]]]]
=> 0
[[[],[]],[],[]]
=> 0
[[[[]]],[],[]]
=> 0
[[[],[]],[[]]]
=> 0
[[[[]]],[[]]]
=> 0
[[[],[],[]],[]]
=> 0
[[[],[[]]],[]]
=> 0
[[[[]],[]],[]]
=> 0
[[[[],[]]],[]]
=> 0
[[[[[]]]],[]]
=> 0
Description
The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.
Matching statistic: St001107
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
St001107: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> 0
[[[]],[]]
=> [1,1,0,0,1,0]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 0
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[[[[]]],[],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[[[[]]],[],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[[[[]]],[],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[[[[]]],[],[[]],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
[[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 0
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 0
[[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 0
[[[],[[],[[],[]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[[[],[[[],[]],[]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 0
[[[[],[]],[[],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 0
[[[[],[[],[]]],[]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 0
[[[[[],[]],[]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0
[[[[[],[[[]]]]]],[]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[[[[[[]]],[]]]],[]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[[[[[[],[],[]]]]],[]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 0
[[[[[[],[[]]]]]],[]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[[[[[[]],[]]]]],[]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0
[[[[[[[],[]]]]]],[]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0
[[[[[[[[]]]]]]],[]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 0
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 0
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 0
[[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
[[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0
[[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 0
[[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 0
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 0
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 0
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 0
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 0
[[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
[[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0
[[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 0
[[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 0
[[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 0
[[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 0
[[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 0
[[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 0
[[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 0
[[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000674
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1,0]
=> ? = 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 0
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 0
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[],[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[],[],[[]]]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[],[[],[]]]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[],[[[]]]]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[]],[],[]]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[]],[[]]]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[],[]],[]]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[[]]],[]]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[],[],[]]]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[],[[]]]]
=> [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[[]],[]]]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[[],[]]]]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 0
[[],[],[],[],[[[[]]]]]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
[[],[],[],[[]],[],[],[]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[]],[],[[]]]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[]],[[]],[]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[]],[[],[]]]
=> [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [7,6,8,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[]],[[[]]]]
=> [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[],[]],[],[]]
=> [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[[]]],[],[]]
=> [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[],[]],[[]]]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ?
=> ? = 0
[[],[],[],[[[]]],[[]]]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 0
[[],[],[],[[],[],[]],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[],[[]]],[]]
=> [.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [8,5,6,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[[]],[]],[]]
=> [.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [8,6,4,5,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[[],[]]],[]]
=> [.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [8,5,4,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[[[]]]],[]]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[],[[],[],[],[]]]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[],[],[[]]]]
=> [.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[],[[]],[]]]
=> [.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [7,5,6,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[],[[],[]]]]
=> [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[],[[[]]]]]
=> [.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [5,6,7,4,8,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[]],[],[]]]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [7,6,4,5,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[]],[[]]]]
=> [.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[],[]],[]]]
=> [.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[[]]],[]]]
=> [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [7,4,5,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[],[],[]]]]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[],[[]]]]]
=> [.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [5,6,4,7,8,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[[]],[]]]]
=> [.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[[],[]]]]]
=> [.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [5,4,6,7,8,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
[[],[],[],[[[[[]]]]]]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
[[],[],[[]],[],[],[],[]]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[[]],[],[],[[]]]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[[]],[],[[]],[]]
=> [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[[]],[],[[],[]]]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[[],[],[[]],[],[[[]]]]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 0
[[],[],[[]],[[]],[],[]]
=> [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[[],[],[[]],[[]],[[]]]
=> [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000993
Mp00246: Ordered trees —rotate⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 75%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 75%
Values
[[]]
=> [[]]
=> [1,0]
=> []
=> ? = 1 + 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> [1]
=> ? = 0 + 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> []
=> ? = 2 + 1
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1 = 0 + 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[[[]],[]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> ? = 0 + 1
[[[],[]]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2 = 1 + 1
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? = 3 + 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[[[]],[],[]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 0 + 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? = 0 + 1
[[[],[],[]]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 1 + 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 4 + 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 0 + 1
[[[],[],[],[]]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2 = 1 + 1
[[[],[],[[]]]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 1 + 1
[[[],[[]],[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2 = 1 + 1
[[[],[[],[]]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2 = 1 + 1
[[[],[[[]]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2 = 1 + 1
[[[[]],[],[]]]
=> [[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[[[[]],[[]]]]
=> [[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2 = 1 + 1
[[[[],[]],[]]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[[[[[[]]]]]]
=> [[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 5 + 1
[[],[],[],[],[],[]]
=> [[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 0 + 1
[[],[],[],[],[[]]]
=> [[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 0 + 1
[[],[],[],[[]],[]]
=> [[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 0 + 1
[[],[],[],[[],[]]]
=> [[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? = 0 + 1
[[],[],[[]],[],[]]
=> [[],[[]],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 0 + 1
[[],[],[[]],[[]]]
=> [[],[[]],[[]],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 0 + 1
[[],[],[[],[]],[]]
=> [[],[[],[]],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? = 0 + 1
[[],[[]],[],[],[]]
=> [[[]],[],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 0 + 1
[[],[[]],[],[[]]]
=> [[[]],[],[[]],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 0 + 1
[[],[[]],[[]],[]]
=> [[[]],[[]],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 0 + 1
[[],[[],[]],[],[]]
=> [[[],[]],[],[],[]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 0 + 1
[[[],[],[],[]],[]]
=> [[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 0 + 1
[[[[[[]]]]],[]]
=> [[[[[[],[]]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? = 0 + 1
[[[],[],[],[],[]]]
=> [[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 1 + 1
[[[],[],[],[[]]]]
=> [[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 1 + 1
[[[],[],[[]],[]]]
=> [[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 1 + 1
[[[],[[]],[],[]]]
=> [[[]],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 1 + 1
[[[[[[[]]]]]]]
=> [[[[[[[]]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 6 + 1
[[],[],[],[],[],[],[]]
=> [[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 0 + 1
[[],[],[],[],[],[[]]]
=> [[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 0 + 1
[[],[],[],[],[[]],[]]
=> [[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 0 + 1
[[],[],[],[],[[],[]]]
=> [[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 0 + 1
[[],[],[],[],[[[]]]]
=> [[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 0 + 1
[[],[],[],[[]],[],[]]
=> [[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 0 + 1
[[],[],[],[[]],[[]]]
=> [[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 0 + 1
[[],[],[],[[],[]],[]]
=> [[],[],[[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 0 + 1
[[],[],[],[[[]]],[]]
=> [[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 0 + 1
[[],[],[],[[],[],[]]]
=> [[],[],[[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 0 + 1
[[],[],[],[[],[[]]]]
=> [[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? = 0 + 1
[[],[],[],[[[]],[]]]
=> [[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 0 + 1
[[],[],[],[[[],[]]]]
=> [[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? = 0 + 1
[[],[],[],[[[[]]]]]
=> [[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 0 + 1
[[],[],[[]],[],[],[]]
=> [[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 0 + 1
[[],[],[[]],[],[[]]]
=> [[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 0 + 1
[[],[],[[]],[[]],[]]
=> [[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 0 + 1
[[],[],[[]],[[],[]]]
=> [[],[[]],[[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 0 + 1
[[],[],[[]],[[[]]]]
=> [[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? = 0 + 1
[[],[],[[],[]],[],[]]
=> [[],[[],[]],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 0 + 1
[[],[],[[[]]],[],[]]
=> [[],[[[]]],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 0 + 1
[[],[],[[],[]],[[]]]
=> [[],[[],[]],[[]],[]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 0 + 1
[[],[],[[[]]],[[]]]
=> [[],[[[]]],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? = 0 + 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000907
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 88%
St000907: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 88%
Values
[[]]
=> ([(0,1)],2)
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 1 = 0 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 1 = 0 + 1
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ? = 0 + 1
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ? = 0 + 1
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ? = 0 + 1
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ? = 0 + 1
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000264
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 17%●distinct values known / distinct values provided: 12%
Values
[[]]
=> [1,0]
=> [1] => ([],1)
=> ? = 1 + 3
[[],[]]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ? = 0 + 3
[[[]]]
=> [1,1,0,0]
=> [2] => ([],2)
=> ? = 2 + 3
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? = 0 + 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> ? = 1 + 3
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => ([],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)
=> 3 = 0 + 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? = 0 + 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? = 0 + 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? = 1 + 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ? = 1 + 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? = 1 + 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 4 + 3
[[],[],[],[],[]]
=> [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)
=> 3 = 0 + 3
[[],[],[],[[]]]
=> [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)
=> 3 = 0 + 3
[[],[],[[]],[]]
=> [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)
=> 3 = 0 + 3
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 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)
=> 3 = 0 + 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[],[[[]]],[]]
=> [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 = 0 + 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 0 + 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 0 + 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 0 + 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 0 + 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 0 + 3
[[[]],[],[],[]]
=> [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)
=> 3 = 0 + 3
[[[]],[],[[]]]
=> [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 = 0 + 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 = 0 + 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[[[]]],[],[]]
=> [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 = 0 + 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> ? = 1 + 3
[[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> ? = 2 + 3
[[[[],[[]]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> ? = 2 + 3
[[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> ? = 2 + 3
[[[[[],[]]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> ? = 3 + 3
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ? = 5 + 3
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[[]]],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[],[[]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[[]],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[],[[[[]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[]],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[]],[[]],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[]],[[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[]],[[[]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[],[]],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[[]]],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[],[]],[[]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[[]]],[[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[],[],[]],[]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[],[[]]],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[[]],[]],[]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[[],[]]],[]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[[[]]]],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> ? = 0 + 3
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? = 0 + 3
[[],[[],[[]],[]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> ? = 0 + 3
[[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? = 0 + 3
[[],[[],[[[]]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? = 0 + 3
[[[]],[],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[[]],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[[]],[],[[]],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001091
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 0
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 2
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 3
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 4
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 0
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000160
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000475
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001933
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[]],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[],[[[[]]]]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[]],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[]],[[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]]],[[]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]]]],[]]
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[],[[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[]],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[],[]]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[],[[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,6),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[]],[]]]
=> ([(0,6),(1,7),(2,7),(3,5),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]]],[]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,5),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,4),(4,6),(5,7),(6,5)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,4)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[],[[[[[]]]]]]
=> ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
[[],[[]],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 0 + 1
Description
The largest multiplicity of a part in an integer partition.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000461The rix statistic of a permutation. St000312The number of leaves in a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001479The number of bridges of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!