Your data matches 29 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000365
(load all 10 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => 0
{{1,2}}
 => [2,1] => 0
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => 1
{{1,2,3,4}}
 => [2,3,4,1] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 0
Description
The number of double ascents of a permutation. A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St000366
(load all 10 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,4,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,4,2,1] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,1,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,4,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,1,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,1,2,3] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [4,3,2,1,5] => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,3,2,5,1] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,3,1,2,5] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,3,5,1,2] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,3,5,2,1] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,2,1,5,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,5,1] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1,2,3,5] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [4,5,2,1,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [4,5,2,3,1] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1,3,2,5] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [4,5,1,2,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [4,5,3,1,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [4,5,3,2,1] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,4,2,1,5] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,1,2,5,4] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,2,5,1,4] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,2,5,4,1] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,4,1,2,5] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,5,2,4] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,4,5,1,2] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [2,3,4,1,5] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,3,1,5,4] => 0
Description
The number of double descents of a permutation. A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000118
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [.,.]
 => [.,.]
 => 0
{{1,2}}
 => [2,1] => [[.,.],.]
 => [[.,.],.]
 => 0
{{1},{2}}
 => [1,2] => [.,[.,.]]
 => [.,[.,.]]
 => 0
{{1,2,3}}
 => [2,3,1] => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => 0
{{1,2},{3}}
 => [2,1,3] => [[.,.],[.,.]]
 => [[.,[.,.]],.]
 => 0
{{1,3},{2}}
 => [3,2,1] => [[[.,.],.],.]
 => [[[.,.],.],.]
 => 0
{{1},{2,3}}
 => [1,3,2] => [.,[[.,.],.]]
 => [.,[[.,.],.]]
 => 0
{{1},{2},{3}}
 => [1,2,3] => [.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => [[.,.],[.,[[.,.],.]]]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
 => [[[.,.],[.,[.,.]]],.]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => [[[.,.],.],[.,[.,.]]]
 => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
 => [[[.,.],[[.,.],.]],.]
 => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],.]
 => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree. [[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St000731
(load all 6 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => [3,2,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [2,3,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,4,2,1] => [2,4,3,1] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [4,3,2,1,5] => [2,3,4,1,5] => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,3,2,5,1] => [5,3,4,2,1] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,3,1,2,5] => [3,1,4,2,5] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,3,5,1,2] => [5,1,4,3,2] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,3,5,2,1] => [2,5,4,3,1] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => [3,4,2,1,5] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,2,1,5,3] => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,5,1] => [5,4,2,3,1] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [4,5,2,1,3] => [2,5,1,4,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [4,5,2,3,1] => [3,5,2,4,1] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1,3,2,5] => [4,3,1,2,5] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [4,5,1,2,3] => [5,1,2,4,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [4,5,3,1,2] => [3,1,5,4,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [4,5,3,2,1] => [2,3,5,4,1] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,4,2,1,5] => [2,4,3,1,5] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,1,2,5,4] => [3,1,2,5,4] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => [5,4,3,2,1] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,2,5,1,4] => [5,3,2,1,4] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,2,5,4,1] => [4,3,2,5,1] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,4,1,2,5] => [4,1,3,2,5] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,5,2,4] => [3,5,1,2,4] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,4,5,1,2] => [5,1,3,4,2] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => [2,5,3,4,1] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [2,3,4,1,5] => [4,2,3,1,5] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,3,1,5,4] => [3,2,1,5,4] => 0
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001066
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [.,.]
 => [1,0]
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2}}
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 0 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000732
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => [3,2,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [3,1,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => [3,1,2,4] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => [2,3,1,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => [2,4,3,1] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,1,3] => [3,1,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,1,2,3] => [2,3,4,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => [2,4,1,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [4,3,2,1,5] => [4,1,2,3,5] => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,3,2,5,1] => [5,4,2,3,1] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,3,1,2,5] => [2,4,1,3,5] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,3,5,1,2] => [2,5,4,3,1] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,3,5,2,1] => [5,1,4,3,2] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => [4,3,1,2,5] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,2,1,5,3] => [4,1,5,2,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,5,1] => [5,3,4,2,1] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,1,2,3,5] => [2,3,4,1,5] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [4,5,2,1,3] => [3,1,5,4,2] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [4,5,2,3,1] => [5,3,1,4,2] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [4,1,3,2,5] => [3,4,2,1,5] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [4,5,1,2,3] => [2,3,5,4,1] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [4,5,3,1,2] => [2,5,1,4,3] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [4,5,3,2,1] => [5,1,2,4,3] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,4,2,1,5] => [4,1,3,2,5] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,1,2,5,4] => [2,3,1,5,4] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => [5,4,3,2,1] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => [3,1,2,4,5] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,2,5,1,4] => [4,3,2,5,1] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,2,5,4,1] => [5,3,2,1,4] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,4,1,2,5] => [2,4,3,1,5] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,5,2,4] => [3,4,1,5,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,4,5,1,2] => [2,5,3,4,1] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => [5,1,3,4,2] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [2,3,4,1,5] => [4,2,3,1,5] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,3,1,5,4] => [3,2,1,5,4] => 0
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [2,3,4,5,1] => [5,2,3,4,1] => 0
Description
The number of double deficiencies of a permutation. A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St000931
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [.,.]
 => [1,0]
 => ? = 0
{{1,2}}
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 0
{{1},{2}}
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 0
{{1,2,3}}
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 0
{{1,2},{3}}
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 0
{{1,3},{2}}
 => [3,2,1] => [[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
{{1},{2,3}}
 => [1,3,2] => [.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 0
{{1},{2},{3}}
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
Description
The number of occurrences of the pattern UUU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000771
(load all 26 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,2] => ([],2)
 => ? = 0 + 1
{{1},{2}}
 => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => ([],3)
 => ? ∊ {0,0} + 1
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2} + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,4,2,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [3,4,2,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,5,4,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3},{5,6}}
 => [2,4,3,1,6,5] => [3,5,4,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [3,2,5,1,4,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [3,2,1,6,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,4,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3},{4},{5,6}}
 => [2,1,3,4,6,5] => [3,2,4,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,3,4},{2},{5,6}}
 => [3,2,4,1,6,5] => [4,3,5,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [4,5,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => [4,5,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,3,5},{2,6},{4}}
 => [3,6,5,4,1,2] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3} + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 26 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => ? = 0 + 1
{{1,2,3}}
 => [2,3,1] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,0} + 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1} + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1} + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,1} + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1} + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1} + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,1} + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2} + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,5},{4},{6}}
 => [2,3,5,4,1,6] => [2,5,3,1,4,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [6,2,1,5,3,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4,5,6},{3}}
 => [2,4,3,5,6,1] => [4,2,3,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3},{5},{6}}
 => [2,4,3,1,5,6] => [2,4,5,3,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,5,6},{3,4}}
 => [2,5,4,3,6,1] => [5,2,4,3,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [2,5,1,4,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2,5},{3,4},{6}}
 => [2,5,4,3,1,6] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001604
(load all 3 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 63%●distinct values known / distinct values provided: 60%
Values
{{1}}
 => [1] => [[1],[]]
 => []
 => ? = 0
{{1,2}}
 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0}
{{1,2,3}}
 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,1}
{{1,2},{3}}
 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1}
{{1,3},{2}}
 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1}
{{1},{2,3}}
 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1}
{{1},{2},{3}}
 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1}
{{1,2,3,4}}
 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,2,3},{4}}
 => [3,1] => [[3,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,2,4},{3}}
 => [3,1] => [[3,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,2},{3,4}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,2},{3},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,3,4},{2}}
 => [3,1] => [[3,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,3},{2,4}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,3},{2},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,4},{2,3}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1},{2,3,4}}
 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1},{2,3},{4}}
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,4},{2},{3}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1},{2,4},{3}}
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1},{2},{3,4}}
 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1},{2},{3},{4}}
 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2}
{{1,2,3,4,5}}
 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2,3,4},{5}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2,3},{4},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2,4},{3},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2},{3,4,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2},{3,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,2},{3},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1,3,4,5},{2}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,3,4},{2},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,3},{2,4,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,3},{2,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,3},{2},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1,4,5},{2,3}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,4},{2,3,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,4},{2,3},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,5},{2,3,4}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,3,4,5}}
 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,3,4},{5}}
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,5},{2,3},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1},{2,3,5},{4}}
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,3},{4,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,4,5},{2},{3}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,4},{2},{3,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1,5},{2,4},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1},{2,4,5},{3}}
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,4},{3,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,5},{2},{3,4}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2,5},{3,4}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2},{3,4,5}}
 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,5},{2},{3},{4}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
{{1,2,3,4,5},{6}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,3,4},{5,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3,4},{5},{6}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,3,5},{4,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3,5},{4},{6}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,3,6},{4},{5}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
{{1,2,4,5,6},{3}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,4,5},{3,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,4,5},{3},{6}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,4,6},{3,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,4},{3,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,4,6},{3},{5}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,4},{3,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,4},{3},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
{{1,2,5,6},{3,4}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,5},{3,4},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2},{3,4,5},{6}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2},{3,4,6},{5}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000260The radius of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001866The nesting alignments of a signed permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001868The number of alignments of type NE of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.