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Your data matches 188 different statistics following compositions of up to 3 maps.
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Matching statistic: St000758
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 2 - 1
[1,1] => 1 = 2 - 1
[2] => 1 = 2 - 1
[1,1,1] => 1 = 2 - 1
[1,2] => 2 = 3 - 1
[2,1] => 1 = 2 - 1
[3] => 1 = 2 - 1
[1,1,1,1] => 1 = 2 - 1
[1,1,2] => 2 = 3 - 1
[1,2,1] => 2 = 3 - 1
[1,3] => 2 = 3 - 1
[2,1,1] => 1 = 2 - 1
[2,2] => 2 = 3 - 1
[3,1] => 1 = 2 - 1
[4] => 1 = 2 - 1
[1,1,1,1,1] => 1 = 2 - 1
[1,1,1,2] => 2 = 3 - 1
[1,1,2,1] => 2 = 3 - 1
[1,1,3] => 2 = 3 - 1
[1,2,1,1] => 2 = 3 - 1
[1,2,2] => 2 = 3 - 1
[1,3,1] => 2 = 3 - 1
[1,4] => 2 = 3 - 1
[2,1,1,1] => 1 = 2 - 1
[2,1,2] => 2 = 3 - 1
[2,2,1] => 2 = 3 - 1
[2,3] => 2 = 3 - 1
[3,1,1] => 1 = 2 - 1
[3,2] => 2 = 3 - 1
[4,1] => 1 = 2 - 1
[5] => 1 = 2 - 1
[2,1,1,1,1] => 1 = 2 - 1
[3,1,1,1] => 1 = 2 - 1
[4,1,1] => 1 = 2 - 1
[5,1] => 1 = 2 - 1
[6] => 1 = 2 - 1
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000396
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [.,.]
=> 1 = 2 - 1
[1,1] => [1,0,1,0]
=> [[.,.],.]
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> [.,[.,.]]
=> 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1 = 2 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 3 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 2 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 2 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 3 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2 = 3 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> 1 = 2 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> 1 = 2 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> 1 = 2 - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 1 = 2 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 1 = 2 - 1
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000862
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1 = 2 - 1
[1,1] => [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[2] => [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 3 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 2 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2 = 3 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 3 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 2 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 1 = 2 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1 = 2 - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1 = 2 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1 = 2 - 1
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St001734
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1 = 2 - 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2] => ([],2)
=> ([],1)
=> 1 = 2 - 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 3 - 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
[3] => ([],3)
=> ([],1)
=> 1 = 2 - 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 3 - 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 3 - 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[4] => ([],4)
=> ([],1)
=> 1 = 2 - 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 3 - 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 3 - 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 3 - 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
[5] => ([],5)
=> ([],1)
=> 1 = 2 - 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 2 - 1
[6] => ([],6)
=> ([],1)
=> 1 = 2 - 1
Description
The lettericity of a graph.
Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$.
The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.
Matching statistic: St000845
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0 = 2 - 2
[1,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 2 - 2
[2] => ([],2)
=> ([],1)
=> 0 = 2 - 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 2 - 2
[1,2] => ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> 0 = 2 - 2
[3] => ([],3)
=> ([],1)
=> 0 = 2 - 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 2 - 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,3] => ([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 2 - 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 2 - 2
[4] => ([],4)
=> ([],1)
=> 0 = 2 - 2
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0 = 2 - 2
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 3 - 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,4] => ([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> 0 = 2 - 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 0 = 2 - 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 2 - 2
[5] => ([],5)
=> ([],1)
=> 0 = 2 - 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> 0 = 2 - 2
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],4)
=> 0 = 2 - 2
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 0 = 2 - 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 2 - 2
[6] => ([],6)
=> ([],1)
=> 0 = 2 - 2
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St001942
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0 = 2 - 2
[1,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 2 - 2
[2] => ([],2)
=> ([],1)
=> 0 = 2 - 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 2 - 2
[1,2] => ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,1] => ([(0,2),(1,2)],3)
=> ([],2)
=> 0 = 2 - 2
[3] => ([],3)
=> ([],1)
=> 0 = 2 - 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 2 - 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,3] => ([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 0 = 2 - 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 0 = 2 - 2
[4] => ([],4)
=> ([],1)
=> 0 = 2 - 2
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0 = 2 - 2
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 3 - 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,4] => ([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> 0 = 2 - 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 0 = 2 - 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0 = 2 - 2
[5] => ([],5)
=> ([],1)
=> 0 = 2 - 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> 0 = 2 - 2
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],4)
=> 0 = 2 - 2
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 0 = 2 - 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> 0 = 2 - 2
[6] => ([],6)
=> ([],1)
=> 0 = 2 - 2
Description
The number of loops of the quiver corresponding to the reduced incidence algebra of a poset.
Matching statistic: St000397
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [.,.]
=> [[],[]]
=> 2
[1,1] => [1,0,1,0]
=> [[.,.],.]
=> [[[],[]],[]]
=> 2
[2] => [1,1,0,0]
=> [.,[.,.]]
=> [[],[[],[]]]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2
[1,2] => [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3
[2,1] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 2
[3] => [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 3
[1,3] => [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> [[[[[[],[[],[]]],[]],[]],[]],[]]
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> [[[[[],[[],[[],[]]]],[]],[]],[]]
=> 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> [[[[],[[],[[],[[],[]]]]],[]],[]]
=> 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [[[],[[],[[],[[],[[],[]]]]]],[]]
=> 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [[],[[],[[],[[],[[],[[],[]]]]]]]
=> 2
Description
The Strahler number of a rooted tree.
Matching statistic: St000470
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [2,1] => [2,1] => 2
[1,1] => [1,0,1,0]
=> [3,1,2] => [2,3,1] => 2
[2] => [1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 2
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 3
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 2
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => 3
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => 2
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,2,3,4,1,5] => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,6,2,1,3,4] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,6,2,1,3,5] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,2,3] => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,6,4,2,3,5] => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,5,6,2,4] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,6,2,5] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,3,4,5,6,2] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,1,3,4,2,5] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,1,2,6,3,4] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,1,2,6,3,5] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,4,5,6,3] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,1,2,4,3,5] => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,5,6,4] => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,6,5] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,3,4,5,6,7,2] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,2,4,5,6,7,3] => 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,2,3,5,6,7,4] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,2,3,4,6,7,5] => 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000092
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1 = 2 - 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1 = 2 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 2 = 3 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 2 = 3 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 2 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1 = 2 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 1 = 2 - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 1 = 2 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 1 = 2 - 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000099
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1 = 2 - 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2 = 3 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 2 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 3 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 2 = 3 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2 = 3 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 2 = 3 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 2 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 1 = 2 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1 = 2 - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1 = 2 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1 = 2 - 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
The following 178 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000183The side length of the Durfee square of an integer partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000453The number of distinct Laplacian eigenvalues of a graph. St000679The pruning number of an ordered tree. St000920The logarithmic height of a Dyck path. St001093The detour number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001735The number of permutations with the same set of runs. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000023The number of inner peaks of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000846The maximal number of elements covering an element of a poset. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001743The discrepancy of a graph. St001840The number of descents of a set partition. St000662The staircase size of the code of a permutation. St000619The number of cyclic descents of a permutation. St001330The hat guessing number of a graph. St000353The number of inner valleys of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001394The genus of a permutation. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000640The rank of the largest boolean interval in a poset. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St000035The number of left outer peaks of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000761The number of ascents in an integer composition. St000805The number of peaks of the associated bargraph. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St000091The descent variation of a composition. St000456The monochromatic index of a connected graph. St000709The number of occurrences of 14-2-3 or 14-3-2. St001868The number of alignments of type NE of a signed permutation. St001964The interval resolution global dimension of a poset. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000181The number of connected components of the Hasse diagram for the poset. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. 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. 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. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. 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. St001763The Hurwitz number of an integer partition. 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. 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. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000934The 2-degree of an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001948The number of augmented double ascents of a permutation.
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