Your data matches 61 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001237
(load all 9 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => 2
[.,[.,.]]
 => [1,1,0,0]
 => 3
[[.,.],.]
 => [1,0,1,0]
 => 3
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 4
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 4
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 4
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 3
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 4
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 5
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 5
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 5
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 5
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 5
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 4
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 5
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 4
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 4
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 4
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 4
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 5
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 6
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 6
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 6
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 6
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 5
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 5
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 5
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St000245
(load all 6 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 2 - 2
[.,[.,.]]
 => [2,1] => [1,2] => 1 = 3 - 2
[[.,.],.]
 => [1,2] => [1,2] => 1 = 3 - 2
[.,[.,[.,.]]]
 => [3,2,1] => [1,3,2] => 1 = 3 - 2
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 2 = 4 - 2
[[.,.],[.,.]]
 => [1,3,2] => [1,2,3] => 2 = 4 - 2
[[.,[.,.]],.]
 => [2,1,3] => [1,2,3] => 2 = 4 - 2
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 2 = 4 - 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,2,3] => 2 = 4 - 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,3,2,4] => 2 = 4 - 2
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => 2 = 4 - 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,4,2] => 2 = 4 - 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 3 = 5 - 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,2,4,3] => 2 = 4 - 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,2,3,4] => 3 = 5 - 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [1,2,3,4] => 3 = 5 - 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3,4] => 3 = 5 - 2
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,3,2,4] => 2 = 4 - 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,3,4] => 3 = 5 - 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,3,4] => 3 = 5 - 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,2,3,4] => 3 = 5 - 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,5,2,4,3] => 2 = 4 - 2
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,4,2,5,3] => 2 = 4 - 2
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,3,4,2,5] => 3 = 5 - 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,4,2,3,5] => 3 = 5 - 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,3,5,2,4] => 3 = 5 - 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,2,5,3,4] => 3 = 5 - 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,2,4,3,5] => 3 = 5 - 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,3,5,2,4] => 3 = 5 - 2
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,2,3,5,4] => 3 = 5 - 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,4,5,2,3] => 3 = 5 - 2
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,3,2,4,5] => 3 = 5 - 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,2,4,5,3] => 3 = 5 - 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,3,4,5,2] => 3 = 5 - 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 6 - 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,2,5,3,4] => 3 = 5 - 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,2,4,3,5] => 3 = 5 - 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,2,3,5,4] => 3 = 5 - 2
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,2,4,5,3] => 3 = 5 - 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,2,3,4,5] => 4 = 6 - 2
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [1,2,3,5,4] => 3 = 5 - 2
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [1,2,3,4,5] => 4 = 6 - 2
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,3,5,4] => 3 = 5 - 2
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,3,4,5] => 4 = 6 - 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [1,3,2,4,5] => 3 = 5 - 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [1,2,3,4,5] => 4 = 6 - 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,2,3,4,5] => 4 = 6 - 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [1,2,3,4,5] => 4 = 6 - 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4,5] => 4 = 6 - 2
Description
The number of ascents of a permutation.
Matching statistic: St001615
(load all 3 compositions to match this statistic)
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
Description
The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
(load all 3 compositions to match this statistic)
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
Description
The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying $$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001622
(load all 3 compositions to match this statistic)
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => 0 = 2 - 2
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 5 - 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000380
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1]
 => 2
[.,[.,.]]
 => [2,1] => [1,2] => [1,1]
 => 3
[[.,.],.]
 => [1,2] => [1,2] => [1,1]
 => 3
[.,[.,[.,.]]]
 => [3,2,1] => [1,3,2] => [2,1]
 => 3
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,1,1]
 => 4
[[.,.],[.,.]]
 => [1,3,2] => [1,2,3] => [1,1,1]
 => 4
[[.,[.,.]],.]
 => [2,1,3] => [1,2,3] => [1,1,1]
 => 4
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,1,1]
 => 4
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,2,3] => [2,1,1]
 => 4
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,3,2,4] => [2,1,1]
 => 4
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 4
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,4,2] => [2,1,1]
 => 4
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 5
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,2,4,3] => [2,1,1]
 => 4
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 5
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 5
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 5
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,3,2,4] => [2,1,1]
 => 4
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 5
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 5
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 5
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 5
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
 => 4
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,4,2,5,3] => [2,2,1]
 => 4
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,3,4,2,5] => [2,1,1,1]
 => 5
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,4,2,3,5] => [2,1,1,1]
 => 5
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,3,5,2,4] => [2,1,1,1]
 => 5
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,2,5,3,4] => [2,1,1,1]
 => 5
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,2,4,3,5] => [2,1,1,1]
 => 5
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,3,5,2,4] => [2,1,1,1]
 => 5
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,4,5,2,3] => [2,1,1,1]
 => 5
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,3,2,4,5] => [2,1,1,1]
 => 5
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,2,4,5,3] => [2,1,1,1]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,3,4,5,2] => [2,1,1,1]
 => 5
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,2,5,3,4] => [2,1,1,1]
 => 5
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,2,4,3,5] => [2,1,1,1]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
 => 5
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,2,4,5,3] => [2,1,1,1]
 => 5
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
 => 5
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
 => 5
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [1,3,2,4,5] => [2,1,1,1]
 => 5
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 6
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St001211
(load all 2 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[.,[.,.]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 6
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 6
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 5
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 6
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 6
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 5
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 6
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 5
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 5
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 6
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 6
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 6
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 5
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 6
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 5
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 5
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 6
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 6
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 5
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 5
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 5
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 5
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 6
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 3 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[.,[.,.]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 3
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 4
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 5
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 5
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 5
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 5
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 5
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 5
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 5
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 5
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 5
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 5
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 4
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 5
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 5
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 5
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 5
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 6
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 5
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 6
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 5
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 6
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 5
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 6
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 6
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 6
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000010
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1]
 => 1 = 2 - 1
[.,[.,.]]
 => [2,1] => [1,2] => [1,1]
 => 2 = 3 - 1
[[.,.],.]
 => [1,2] => [1,2] => [1,1]
 => 2 = 3 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,3,2] => [2,1]
 => 2 = 3 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[[.,[.,.]],.]
 => [2,1,3] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,1,1]
 => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,2,3] => [2,1,1]
 => 3 = 4 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,3,2,4] => [2,1,1]
 => 3 = 4 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,4,2] => [2,1,1]
 => 3 = 4 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,2,4,3] => [2,1,1]
 => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,3,2,4] => [2,1,1]
 => 3 = 4 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
 => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,4,2,5,3] => [2,2,1]
 => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,3,4,2,5] => [2,1,1,1]
 => 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,4,2,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,3,5,2,4] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,2,5,3,4] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,2,4,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,3,5,2,4] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,4,5,2,3] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,3,2,4,5] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,2,4,5,3] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,3,4,5,2] => [2,1,1,1]
 => 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,2,5,3,4] => [2,1,1,1]
 => 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,2,4,3,5] => [2,1,1,1]
 => 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
 => 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,2,4,5,3] => [2,1,1,1]
 => 4 = 5 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
 => 4 = 5 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
 => 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [1,3,2,4,5] => [2,1,1,1]
 => 4 = 5 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 6 - 1
Description
The length of the partition.
Matching statistic: St000213
(load all 2 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 1 = 2 - 1
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 2 = 3 - 1
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 2 = 3 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[[.,[.,.]],.]
 => [2,1,3] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3 = 4 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 3 = 4 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,3,2] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,4,2,5,3] => [1,4,5,2,3] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,3,4,2,5] => [1,4,3,2,5] => 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,3,5,2,4] => [1,4,3,5,2] => 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,3,5,2,4] => [1,4,3,5,2] => 4 = 5 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,4,5,2,3] => [1,3,5,4,2] => 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4 = 5 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,3,4,5,2] => [1,5,3,4,2] => 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 4 = 5 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as $$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000784The maximum of the length and the largest part of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000021The number of descents of a permutation. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St001176The size of a partition minus its first part. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001812The biclique partition number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St001626The number of maximal proper sublattices of a lattice. St001613The binary logarithm of the size of the center of a lattice. St000907The number of maximal antichains of minimal length in a poset. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001960The number of descents of a permutation minus one if its first entry is not one. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000216The absolute length of a permutation. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000442The maximal area to the right of an up step of a Dyck path. St001935The number of ascents in a parking function. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001875The number of simple modules with projective dimension at most 1.