- FindStat

Your data matches 89 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000196
(load all 11 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => 0
[1,2] => [.,[.,.]]
 => 0
[2,1] => [[.,.],.]
 => 0
[1,2,3] => [.,[.,[.,.]]]
 => 0
[1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [[.,.],[.,.]]
 => 1
[2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [[.,.],[.,.]]
 => 1
[3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 1

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by

$2^{n-2}$ .

Matching statistic: St000201
(load all 11 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
 => 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 2 = 1 + 1

Description

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size

$n$ , at least

$1$ , with exactly one leaf node for is

$2^{n-1}$ , see [2]. The number of binary tree of size

$n$ , at least

$3$ , with exactly two leaf nodes is

$n(n+1)2^{n-2}$ , see [3].

Matching statistic: St000386
(load all 13 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => [1,0]
 => 0
[1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 0
[2,1] => [[.,.],.]
 => [1,1,0,0]
 => 0
[1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 0
[2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 0
[3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1

Description

The number of factors DDU in a Dyck path.

Matching statistic: St000632
(load all 2 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => ([],1)
 => 0
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 0
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 0
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0
[3,1,2] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1

Description

The jump number of the poset. A jump in a linear extension

$e_1, \dots, e_n$ of a poset

$P$ is a pair

$(e_i, e_{i+1})$ so that

$e_{i+1}$ does not cover

$e_i$ in

$P$ . The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Matching statistic: St001037
(load all 3 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000068

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => ([],1)
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1

Description

The number of minimal elements in a poset.

Matching statistic: St000071
(load all 2 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => ([],1)
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1

Description

The number of maximal chains in a poset.

Matching statistic: St000527
(load all 2 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => ([],1)
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2 = 1 + 1

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St000257
(load all 2 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => []
 => 0
[1,2] => [2] => [1,1,0,0]
 => []
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => []
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 0
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 0
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1

Description

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

Matching statistic: St000272

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1]]
 => [1] => ([],1)
 => 0
[1,2] => [[1,2]]
 => [2] => ([],2)
 => 0
[2,1] => [[1],[2]]
 => [2] => ([],2)
 => 0
[1,2,3] => [[1,2,3]]
 => [3] => ([],3)
 => 0
[1,3,2] => [[1,2],[3]]
 => [3] => ([],3)
 => 0
[2,1,3] => [[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => [[1,2],[3]]
 => [3] => ([],3)
 => 0
[3,1,2] => [[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [[1],[2],[3]]
 => [3] => ([],3)
 => 0
[1,2,3,4] => [[1,2,3,4]]
 => [4] => ([],4)
 => 0
[1,2,4,3] => [[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[1,3,2,4] => [[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,4,2] => [[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[1,4,2,3] => [[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[2,1,3,4] => [[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,1,4,3] => [[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,3,1,4] => [[1,2,4],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,3,4,1] => [[1,2,3],[4]]
 => [4] => ([],4)
 => 0
[2,4,1,3] => [[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,3,1] => [[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[3,1,2,4] => [[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => [[1,3],[2,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,4,2,1] => [[1,2],[3],[4]]
 => [4] => ([],4)
 => 0
[4,1,2,3] => [[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[4,1,3,2] => [[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [[1,3],[2],[4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,3,2,1] => [[1],[2],[3],[4]]
 => [4] => ([],4)
 => 0
[1,2,3,4,5] => [[1,2,3,4,5]]
 => [5] => ([],5)
 => 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => [5] => ([],5)
 => 0
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
 => [5] => ([],5)
 => 0
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000292The number of ascents of a binary word. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St000010The length of the partition. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000528The height of a poset. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001732The number of peaks visible from the left. St001963The tree-depth of a graph. St000291The number of descents of a binary word. St000568The hook number of a binary tree. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000806The semiperimeter of the associated bargraph. St000080The rank of the poset. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000353The number of inner valleys of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001668The number of points of the poset minus the width of the poset. St000455The second largest eigenvalue of a graph if it is integral. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000092The number of outer peaks of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000523The number of 2-protected nodes of a rooted tree. St000646The number of big ascents of a permutation. St001812The biclique partition number of a graph. St000409The number of pitchforks in a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001597The Frobenius rank of a skew partition. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001487The number of inner corners of a skew partition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001823The Stasinski-Voll length of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.