searching the database
Your data matches 583 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001512
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St001512: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001512: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 2
[.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 2
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
Description
The minimum rank of a graph.
The minimum rank of a simple graph G is the smallest possible rank over all symmetric real matrices whose entry in row $i$ and column $j$ (for $i\neq j$) is nonzero whenever $\{i, j\}$ is an edge in
$G$, and zero otherwise.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [2]
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 2
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 2
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 2
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 5
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001615
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
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 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
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: St000459
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000870
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001318
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Matching statistic: St001321
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 5 + 1
Description
The number of vertices of the largest induced subforest of a graph.
Matching statistic: St001384
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [2]
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 2 = 1 + 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3 = 2 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5 = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6 = 5 + 1
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
The following 573 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001571The Cartan determinant of the integer partition. St000245The number of ascents of a permutation. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000519The largest length of a factor maximising the subword complexity. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000778The metric dimension of a graph. St001176The size of a partition minus its first part. St001298The number of repeated entries in the Lehmer code of a permutation. St001949The rigidity index of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001554The number of distinct nonempty subtrees of a binary tree. St001883The mutual visibility number of a graph. St000543The size of the conjugacy class of a binary word. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000393The number of strictly increasing runs in a binary word. St001120The length of a longest path in a graph. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001391The disjunction number of a graph. St000806The semiperimeter of the associated bargraph. St001875The number of simple modules with projective dimension at most 1. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001846The number of elements which do not have a complement in the lattice. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000528The height of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001820The size of the image of the pop stack sorting operator. St000070The number of antichains in a poset. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000108The number of partitions contained in the given partition. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000672The number of minimal elements in Bruhat order not less than the permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001096The size of the overlap set of a permutation. St001644The dimension of a graph. St001651The Frankl number of a lattice. St000022The number of fixed points of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000153The number of adjacent cycles of a permutation. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001052The length of the exterior of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000914The sum of the values of the Möbius function of a poset. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000837The number of ascents of distance 2 of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000050The depth or height of a binary tree. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001245The cyclic maximal difference between two consecutive entries of a permutation. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000505The biggest entry in the block containing the 1. St000019The cardinality of the support of a permutation. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St001958The degree of the polynomial interpolating the values of a permutation. St001430The number of positive entries in a signed permutation. St000863The length of the first row of the shifted shape of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St000460The hook length of the last cell along the main diagonal of an integer partition. St000080The rank of the poset. St001649The length of a longest trail in a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000189The number of elements in the poset. St000299The number of nonisomorphic vertex-induced subtrees. St001717The largest size of an interval in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000167The number of leaves of an ordered tree. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001479The number of bridges of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001623The number of doubly irreducible elements of a lattice. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000081The number of edges of a graph. St000087The number of induced subgraphs. St000171The degree of the graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000625The sum of the minimal distances to a greater element. St000632The jump number of the poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St001093The detour number of a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001322The size of a minimal independent dominating set in a graph. St001530The depth of a Dyck path. St001828The Euler characteristic of a graph. St000026The position of the first return of a Dyck path. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000312The number of leaves in a graph. St000527The width of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001626The number of maximal proper sublattices of a lattice. St000209Maximum difference of elements in cycles. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000448The number of pairs of vertices of a graph with distance 2. St000741The Colin de Verdière graph invariant. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001323The independence gap of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St001962The proper pathwidth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000117The number of centered tunnels of a Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000271The chromatic index of a graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000385The number of vertices with out-degree 1 in a binary tree. St000443The number of long tunnels of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000703The number of deficiencies of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001118The acyclic chromatic index of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001286The annihilation number of a graph. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001481The minimal height of a peak of a Dyck path. St001725The harmonious chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001963The tree-depth of a graph. St000010The length of the partition. St000058The order of a permutation. St000147The largest part of an integer partition. St000301The number of facets of the stable set polytope of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000468The Hosoya index of a graph. St000722The number of different neighbourhoods in a graph. St000915The Ore degree of a graph. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001389The number of partitions of the same length below the given integer partition. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001492The 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. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000656The number of cuts of a poset. 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. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000060The greater neighbor of the maximum. St000197The number of entries equal to positive one in the alternating sign matrix. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001480The number of simple summands of the module J^2/J^3. St000064The number of one-box pattern of a permutation. St000836The number of descents of distance 2 of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St001959The product of the heights of the peaks of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St000018The number of inversions of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000924The number of topologically connected components of a perfect matching. St001268The size of the largest ordinal summand in the poset. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000699The toughness times the least common multiple of 1,. St000095The number of triangles of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000450The number of edges minus the number of vertices plus 2 of a graph. St000673The number of non-fixed points of a permutation. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001834The number of non-isomorphic minors of a graph. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001468The smallest fixpoint of a permutation. St001706The number of closed sets in a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001091The number of parts in an integer partition whose next smaller part has the same size. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000160The multiplicity of the smallest part of a partition. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000547The number of even non-empty partial sums of an integer partition. St000809The reduced reflection length of the permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001117The game chromatic index of a graph. St001429The number of negative entries in a signed permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000086The number of subgraphs. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001128The exponens consonantiae of a partition. St000456The monochromatic index of a connected graph. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001948The number of augmented double ascents of a permutation. St001060The distinguishing index of a graph. St001557The number of inversions of the second entry of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001927Sparre Andersen's number of positives of a signed permutation. St000636The hull number of a graph. St000918The 2-limited packing number of a graph. St001029The size of the core of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001642The Prague dimension of a graph. St001656The monophonic position number of a graph. St001734The lettericity of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000260The radius of a connected graph. St000535The rank-width of a graph. St001395The number of strictly unfriendly partitions of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001792The arboricity of a graph. St000201The number of leaf nodes in a binary tree. St001434The number of negative sum pairs of a signed permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000834The number of right outer peaks of a permutation. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000509The diagonal index (content) of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000937The number of positive values of the symmetric group character corresponding to the partition. St000264The girth of a graph, which is not a tree. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000997The even-odd crank of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000477The weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000284The Plancherel distribution on integer partitions. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001943The sum of the squares of the hook lengths of an integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000884The number of isolated descents of a permutation. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001817The number of flag weak exceedances of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001892The flag excedance statistic of a signed permutation. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001248Sum of the even parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001657The number of twos in an integer partition. St001763The Hurwitz number of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000137The Grundy value of an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001281The normalized isoperimetric number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001545The second Elser number of a connected graph. St000135The number of lucky cars of the parking function. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St000442The maximal area to the right of an up step of a Dyck path. St000919The number of maximal left branches of a binary tree. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001613The binary logarithm of the size of the center of a lattice. St000148The number of odd parts of a partition. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000568The hook number of a binary tree. St000691The number of changes of a binary word. St001214The aft of an integer partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001437The flex of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001555The order of a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!