- FindStat

Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000054
(load all 5 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation

$\pi$ of

$n$ , together with its rotations, obtained by conjugating with the long cycle

$(1,\dots,n)$ . Drawing the labels

$1$ to

$n$ in this order on a circle, and the arcs

$(i, \pi(i))$ as straight lines, the rotation of

$\pi$ is obtained by replacing each number

$i$ by

$(i\bmod n) +1$ . Then,

$\pi(1)-1$ is the number of rotations of

$\pi$ where the arc

$(1, \pi(1))$ is a deficiency. In particular, if

$O(\pi)$ is the orbit of rotations of

$\pi$ , then the number of deficiencies of

$\pi$ equals

$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$

Matching statistic: St001640
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

Matching statistic: St000007
(load all 10 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000011
(load all 5 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000025
(load all 3 compositions to match this statistic)

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000068

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in a poset.

Matching statistic: St000740
(load all 7 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last entry of a permutation. This statistic is undefined for the empty permutation.

Matching statistic: St000971

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The smallest closer of a set partition. A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers. In other words, this is the smallest among the maximal elements of the blocks.

Matching statistic: St001050

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.

Matching statistic: St001784
(load all 2 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001784: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimum of the smallest closer and the second element of the block containing 1 in a set partition. A closer of a set partition is the maximal element of a non-singleton block. This statistic is defined as

$1$ if

$\{1\}$ is a singleton block, and otherwise the minimum of the smallest closer and the second element of the block containing

$1$ .

The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000439The position of the first down step of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000133The "bounce" of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000005The bounce statistic of a Dyck path. St000051The size of the left subtree of a binary tree. St000120The number of left tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000947The major index east count of a Dyck path. St000989The number of final rises of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000738The first entry in the last row of a standard tableau. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St000260The radius of a connected graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000383The last part of an integer composition. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.