- FindStat

Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000022
(load all 21 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000221
(load all 13 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong fixed points of a permutation. $i$ is called a strong fixed point of $\pi$ if 1. $j < i$ implies $\pi_j < \pi_i$, and 2. $j > i$ implies $\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St000117

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of centered tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.

Matching statistic: St000241
(load all 8 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.

Matching statistic: St000315

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated vertices of a graph.

Matching statistic: St000445

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St000475
(load all 4 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000895
(load all 5 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ones on the main diagonal of an alternating sign matrix.

Matching statistic: St001008
(load all 3 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000674
(load all 3 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.

The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000247The number of singleton blocks of a set partition. St000894The trace of an alternating sign matrix. St001903The number of fixed points of a parking function.