- FindStat

Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000374
(load all 33 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000996
(load all 29 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St001280
(load all 2 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000291

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => 1 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [2] => 10 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => 11 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => 100 => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => 101 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,2] => 110 => 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [3] => 100 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => 1000 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => 1000 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 10001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 10010 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => 10000 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => 10000 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => 10000 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => 10001 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => 10000 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => 10000 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 10001 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => 10000 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => 10001 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 10010 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [4,1,1,1,1,1,1] => 1000111111 => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[]
 => []
 => [] => ? => ? = 0
[1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? => ? => ? = 1
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? => ? => ? = 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 0
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
 => [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,9] => 1100000000 => ? = 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [4,1,1,1,1,1,1] => 1000111111 => ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1
[1,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? => ? => ? = 1
[1,1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? => ? => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0,0]
 => ? => ? => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [2,8] => 1010000000 => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,1,0,0]
 => ? => ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,2,2,4] => 1010101000 => ? = 4
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => ? => ? => ? = 1
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? => ? => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2,2] => 1000101010 => ? = 4

Description

The number of descents of a binary word.

Matching statistic: St000390
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] =>  =>  => ? = 0
[1,0,1,0]
 => [2,1] => 0 => 1 => 1
[1,1,0,0]
 => [1,2] => 1 => 0 => 0
[1,0,1,0,1,0]
 => [2,3,1] => 00 => 11 => 1
[1,0,1,1,0,0]
 => [2,1,3] => 01 => 10 => 1
[1,1,0,0,1,0]
 => [1,3,2] => 10 => 01 => 1
[1,1,0,1,0,0]
 => [3,1,2] => 00 => 11 => 1
[1,1,1,0,0,0]
 => [1,2,3] => 11 => 00 => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 000 => 111 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 001 => 110 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 010 => 101 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 000 => 111 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 011 => 100 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 100 => 011 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 101 => 010 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 000 => 111 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 000 => 111 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 001 => 110 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 110 => 001 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 100 => 011 => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 000 => 111 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 111 => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 0000 => 1111 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 0001 => 1110 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 0010 => 1101 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0000 => 1111 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 0011 => 1100 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 0100 => 1011 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 0101 => 1010 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => 0000 => 1111 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 0000 => 1111 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 0001 => 1110 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 0110 => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 0100 => 1011 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0000 => 1111 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 0111 => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 1000 => 0111 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1001 => 0110 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 1010 => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => 1000 => 0111 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1011 => 0100 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => 0000 => 1111 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => 0001 => 1110 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => 0000 => 1111 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 0000 => 1111 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => 0001 => 1110 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => 0010 => 1101 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => 0000 => 1111 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => 0000 => 1111 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => 0011 => 1100 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1100 => 0011 => 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1,5,8,4,6,7] => ? => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,3,1,7,8,4,5,6] => ? => ? => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,3,6,8,5,7] => ? => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,1,4,6,7,8,3,5] => ? => ? => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,3,5,8,7] => ? => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [2,1,4,6,8,3,5,7] => ? => ? => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,1,4,7,8,3,5,6] => ? => ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,4,5,6,1,3,8,7] => ? => ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,4,5,6,1,8,3,7] => ? => ? => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,4,5,1,3,8,6,7] => ? => ? => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,6,1,7,8,3,5] => ? => ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5,8] => ? => ? => ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,4,7,1,8,3,5,6] => ? => ? => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [2,1,5,6,3,8,4,7] => ? => ? => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [2,1,5,6,8,3,4,7] => ? => ? => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4,8] => ? => ? => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [2,5,6,1,3,4,8,7] => ? => ? => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [2,5,1,3,4,8,6,7] => ? => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [2,1,6,7,3,8,4,5] => ? => ? => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [2,6,7,1,8,3,4,5] => ? => ? => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [2,1,7,3,8,4,5,6] => ? => ? => ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,6,7,2,8,4,5] => ? => ? => ? = 1
[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,6,2,8,4,5,7] => ? => ? => ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [3,4,5,1,2,8,6,7] => ? => ? => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [3,4,1,2,6,8,5,7] => ? => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,6,1,7,2,5,8] => ? => ? => ? = 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [3,4,6,1,8,2,5,7] => ? => ? => ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [3,4,1,7,8,2,5,6] => ? => ? => ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [3,4,1,2,8,5,6,7] => ? => ? => ? = 2
[1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [3,1,2,5,7,8,4,6] => ? => ? => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,5,6,1,7,2,4,8] => ? => ? => ? = 1
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,5,6,7,1,2,4,8] => ? => ? => ? = 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [3,5,6,1,2,4,8,7] => ? => ? => ? = 2
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [3,1,2,6,8,4,5,7] => ? => ? => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [3,6,7,1,8,2,4,5] => ? => ? => ? = 1
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [3,6,8,1,2,4,5,7] => ? => ? => ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [3,1,2,7,8,4,5,6] => ? => ? => ? = 2
[1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ? => ? => ? = 1
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,4,6,7,8,2,3,5] => ? => ? => ? = 1
[1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,4,6,2,8,3,5,7] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
 => [4,1,5,6,2,8,3,7] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
 => [4,1,5,6,2,3,7,8] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
 => [4,1,5,7,2,3,6,8] => ? => ? => ? = 1
[1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
 => [4,1,5,2,3,6,7,8] => ? => ? => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3,8] => ? => ? => ? = 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
 => [4,5,6,1,2,8,3,7] => ? => ? => ? = 1
[1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
 => [4,5,1,2,3,7,8,6] => ? => ? => ? = 2
[1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => [4,5,1,7,2,3,6,8] => ? => ? => ? = 1
[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6,8] => ? => ? => ? = 1

Description

The number of runs of ones in a binary word.

Matching statistic: St000659

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 71%

Values

[1,0]
 => [1] => [1] => [1,0]
 => ? = 0
[1,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,4,1,7,5,8,6] => [4,3,2,1,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1,5,8,4,6,7] => ? => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,1,6,7,8,4,5] => [3,2,1,8,4,7,6,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,6,1,7,8,4,5] => [8,3,2,1,4,7,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,1,4,5,7,8,6] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,3,1,7,8,4,5,6] => [3,2,1,8,4,7,5,6] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,1,8,4,5,6,7] => [3,2,1,8,4,5,6,7] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,7,1,3,8] => [7,2,1,6,5,4,3,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,4,5,6,1,3,8,7] => [6,2,1,5,4,3,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,4,5,6,1,8,3,7] => [8,6,2,1,5,4,3,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3,7,8] => [6,2,1,5,4,3,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,5,1,3,7,8,6] => [5,2,1,4,3,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,4,5,1,3,8,6,7] => ? => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,5,1,3,6,7,8] => [5,2,1,4,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [2,4,1,3,6,5,8,7] => [4,2,1,3,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [2,4,1,3,6,8,5,7] => [4,2,1,3,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,1,6,7,8,3,5] => [8,4,2,1,3,7,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [2,4,1,6,3,5,8,7] => [6,4,2,1,3,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,8,5,7] => [8,6,4,2,1,3,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,1,6,8,3,5,7] => [8,4,2,1,3,6,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,6,1,7,8,3,5] => [8,2,1,4,3,7,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [2,4,6,7,8,1,3,5] => [8,2,1,7,4,3,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5,8] => [7,2,1,6,4,3,5,8] => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [2,4,6,1,3,5,8,7] => [6,2,1,4,3,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,4,6,1,3,8,5,7] => [8,6,2,1,4,3,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [2,4,6,1,8,3,5,7] => [8,2,1,4,3,6,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [2,4,6,8,1,3,5,7] => [8,2,1,6,4,3,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,4,1,3,7,8,5,6] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [2,4,1,7,8,3,5,6] => [8,4,2,1,3,7,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,4,7,1,8,3,5,6] => [8,2,1,4,3,7,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,4,1,3,8,5,6,7] => [4,2,1,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => [4,2,1,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [2,1,5,6,3,8,4,7] => ? => ?
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [2,1,5,6,8,3,4,7] => [2,1,8,3,6,5,4,7] => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [2,5,6,7,8,1,3,4] => [8,2,1,7,3,6,5,4] => ?
 => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4,8] => ? => ?
 => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [2,5,6,1,3,4,8,7] => [6,2,1,5,3,4,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [2,5,6,1,3,8,4,7] => [8,6,2,1,5,3,4,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [2,5,7,8,1,3,4,6] => [8,2,1,7,3,5,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [2,5,1,3,4,8,6,7] => ? => ?
 => ? = 2
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [2,5,1,3,4,6,7,8] => [5,2,1,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [2,1,6,3,4,8,5,7] => [2,1,8,6,3,4,5,7] => ?
 => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [2,1,6,3,8,4,5,7] => [2,1,8,3,4,6,5,7] => ?
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [2,6,1,7,3,4,5,8] => [7,2,1,3,6,4,5,8] => ?
 => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [2,6,7,1,8,3,4,5] => [8,2,1,7,3,4,6,5] => ?
 => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [2,6,7,8,1,3,4,5] => [8,2,1,7,3,6,4,5] => ?
 => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [2,1,7,3,8,4,5,6] => [2,1,8,3,4,7,5,6] => ?
 => ? = 2

Description

The number of rises of length at least 2 of a Dyck path.

Matching statistic: St001949

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)
 => ?
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,4,1,7,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6)],8)
 => ?
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1,5,8,4,6,7] => ?
 => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,1,6,7,8,4,5] => ([(0,5),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7)],8)
 => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,6,1,7,8,4,5] => ([(0,7),(1,7),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
 => ?
 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,1,4,5,7,8,6] => ([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,3,1,7,8,4,5,6] => ([(0,5),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7)],8)
 => ?
 => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,1,8,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7,8] => ([(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,3,6,8,5,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,1,4,6,7,8,3,5] => ([(0,1),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,3,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [2,1,4,6,3,8,5,7] => ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [2,1,4,6,8,3,5,7] => ([(0,1),(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,1,4,7,8,3,5,6] => ([(0,1),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,4,3,8,5,6,7] => ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,7,1,3,8] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,4,5,6,1,3,8,7] => ([(0,1),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,4,5,6,1,8,3,7] => ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3,7,8] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,5,1,3,7,8,6] => ([(0,6),(1,6),(2,7),(3,4),(3,5),(4,7),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,4,5,1,3,8,6,7] => ?
 => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,5,1,3,6,7,8] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [2,4,1,3,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [2,4,1,3,6,8,5,7] => ([(0,7),(1,5),(2,4),(3,6),(4,5),(6,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,1,6,7,8,3,5] => ([(0,5),(1,5),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [2,4,1,6,3,5,8,7] => ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,1,6,8,3,5,7] => ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,6,1,7,8,3,5] => ?
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [2,4,6,7,8,1,3,5] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5,8] => ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [2,4,6,1,3,5,8,7] => ([(0,1),(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,4,6,1,3,8,5,7] => ([(0,7),(1,5),(2,3),(2,7),(3,6),(4,5),(4,6),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [2,4,6,1,8,3,5,7] => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [2,4,6,8,1,3,5,7] => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,4,1,3,7,8,5,6] => ([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(6,7)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [2,4,1,7,8,3,5,6] => ([(0,4),(1,4),(1,7),(2,5),(2,6),(3,5),(3,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,4,7,1,8,3,5,6] => ([(0,6),(1,5),(1,6),(2,4),(2,7),(3,4),(3,7),(4,5),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,4,1,3,8,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6)],8)
 => ?
 => ? = 2
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ([(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [2,1,5,6,7,8,3,4] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [2,1,5,6,3,8,4,7] => ?
 => ?
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [2,1,5,6,8,3,4,7] => ([(0,1),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [2,1,5,7,8,3,4,6] => ([(0,1),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [2,5,6,7,8,1,3,4] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4,8] => ?
 => ?
 => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [2,5,6,1,3,4,8,7] => ([(0,1),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [2,5,6,1,3,8,4,7] => ([(0,7),(1,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 1

Description

The rigidity index of a graph. A base of a permutation group is a set

$B$ such that the pointwise stabilizer of

$B$ is trivial. For example, a base of the symmetric group on

$n$ letters must contain all but one letter. This statistic yields the minimal size of a base for the automorphism group of a graph.

Matching statistic: St001011
(load all 14 compositions to match this statistic)

Mapping

St001011: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 57%

Values

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

Description

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001333
(load all 6 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 57%

Values

[1,0]
 => [1] => [1] => ([],1)
 => 0
[1,0,1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [1,2] => [1,2] => ([],2)
 => 0
[1,0,1,0,1,0]
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => [4,1,2,3,8,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,4,1,7,5,8,6] => [4,1,2,3,8,6,7,5] => ([(0,5),(1,5),(2,5),(3,6),(3,7),(4,6),(4,7),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1,5,8,4,6,7] => ? => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,1,6,7,8,4,5] => [3,1,2,5,8,4,6,7] => ([(0,7),(1,7),(2,5),(3,5),(4,6),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,6,1,7,8,4,5] => [5,8,4,6,1,2,3,7] => ([(0,7),(1,3),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,1,4,5,7,8,6] => [3,1,2,4,5,8,6,7] => ([(2,7),(3,7),(4,6),(5,6)],8)
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,3,1,7,8,4,5,6] => ? => ?
 => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,1,8,4,5,6,7] => ? => ?
 => ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7,8] => ? => ?
 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,3,6,8,5,7] => [2,1,4,3,7,8,5,6] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,1,4,6,7,8,3,5] => ? => ?
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,3,5,8,7] => [2,1,5,6,3,4,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [2,1,4,6,3,8,5,7] => [2,1,7,8,5,6,3,4] => ([(0,1),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [2,1,4,6,8,3,5,7] => [2,1,5,7,8,3,4,6] => ([(0,1),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,1,4,3,7,8,5,6] => [2,1,4,3,6,8,5,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,1,4,7,8,3,5,6] => ? => ?
 => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,4,3,8,5,6,7] => [2,1,4,3,6,7,8,5] => ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,7,1,3,8] => ? => ?
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,4,5,6,1,3,8,7] => ? => ?
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,4,5,6,1,8,3,7] => ? => ?
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3,7,8] => ? => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,5,1,3,7,8,6] => ? => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,4,5,1,3,8,6,7] => ? => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,5,1,3,6,7,8] => ? => ?
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [2,4,1,3,6,5,8,7] => [3,4,1,2,6,5,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [2,4,1,3,6,8,5,7] => [3,4,1,2,7,8,5,6] => ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,1,6,7,8,3,5] => [5,8,3,4,1,2,6,7] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [2,4,1,6,3,5,8,7] => [5,6,3,4,1,2,8,7] => ([(0,1),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,8,5,7] => [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,1,6,8,3,5,7] => [5,7,8,3,4,1,2,6] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,6,1,7,8,3,5] => ? => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [2,4,6,7,8,1,3,5] => ? => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5,8] => ? => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [2,4,6,1,3,5,8,7] => [3,5,6,1,2,4,8,7] => ([(0,1),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,4,6,1,3,8,5,7] => [3,7,8,5,6,1,2,4] => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [2,4,6,1,8,3,5,7] => [5,7,8,3,6,1,2,4] => ?
 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [2,4,6,8,1,3,5,7] => [3,5,7,8,1,2,4,6] => ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,4,1,3,7,8,5,6] => [3,4,1,2,6,8,5,7] => ([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [2,4,1,7,8,3,5,6] => ? => ?
 => ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,4,7,1,8,3,5,6] => ? => ?
 => ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,4,1,3,8,5,6,7] => [3,4,1,2,6,7,8,5] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6)],8)
 => ? = 2
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,4,1,3,5,6,7,8] => ? => ?
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [2,1,5,6,7,8,3,4] => [2,1,4,8,3,5,6,7] => ([(0,1),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [2,1,5,6,3,4,8,7] => [2,1,4,6,3,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [2,1,5,6,3,8,4,7] => ? => ?
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [2,1,5,6,8,3,4,7] => ? => ?
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [2,1,5,7,8,3,4,6] => [2,1,4,6,8,3,5,7] => ([(0,1),(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [2,5,6,7,8,1,3,4] => [3,4,8,1,2,5,6,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4,8] => ? => ?
 => ? = 1

Description

The cardinality of a minimal edge-isolating set of a graph. Let

$\mathcal F$ be a set of graphs. A set of vertices

$S$ is

$\mathcal F$ -isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of

$S$ does not contain any graph in

$\mathcal F$ . This statistic returns the cardinality of the smallest isolating set when

$\mathcal F$ contains only the graph with one edge.

Matching statistic: St001665
(load all 26 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 57%

Values

[1,0]
 => [1] => {{1}}
 => [1] => 0
[1,0,1,0]
 => [2,1] => {{1,2}}
 => [2,1] => 1
[1,1,0,0]
 => [1,2] => {{1},{2}}
 => [1,2] => 0
[1,0,1,0,1,0]
 => [2,3,1] => {{1,2,3}}
 => [2,3,1] => 1
[1,0,1,1,0,0]
 => [2,1,3] => {{1,2},{3}}
 => [2,1,3] => 1
[1,1,0,0,1,0]
 => [1,3,2] => {{1},{2,3}}
 => [1,3,2] => 1
[1,1,0,1,0,0]
 => [3,1,2] => {{1,2,3}}
 => [2,3,1] => 1
[1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => {{1,2,3},{4}}
 => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => {{1},{2,3,4}}
 => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => {{1,2},{3,4,5}}
 => [2,1,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,6,1,4,7] => {{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,6,1,4,5,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
 => [2,3,1,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,6,7,3,5] => {{1,2},{3,4,6},{5,7}}
 => [2,1,4,6,7,3,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,1,4,7,3,5,6] => {{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,4,5,1,6,3,7] => {{1,2,4},{3,5,6},{7}}
 => [2,4,5,1,6,3,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,7,1,3] => {{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,1,3,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,4,5,1,3,7,6] => {{1,2,4},{3,5},{6,7}}
 => [2,4,5,1,3,7,6] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,4,5,7,1,3,6] => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,4,5,1,3,6,7] => {{1,2,4},{3,5},{6},{7}}
 => [2,4,5,1,3,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,6,7,1,3,5] => {{1,2,4,5,7},{3,6}}
 => [2,4,6,5,7,3,1] => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5,7] => {{1,2,4},{3,5,6},{7}}
 => [2,4,5,1,6,3,7] => ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => {{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,1,5,6,7,3,4] => {{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,1,5,7,3,4,6] => {{1,2},{3,5},{4,6,7}}
 => [2,1,5,6,3,7,4] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [2,5,1,3,6,4,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [2,5,1,6,3,4,7] => {{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [2,5,6,1,7,3,4] => {{1,2,4,5,7},{3,6}}
 => [2,4,6,5,7,3,1] => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [2,5,6,7,1,3,4] => {{1,2,5},{3,6},{4,7}}
 => [2,5,6,7,1,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,1,3,4,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [2,5,1,3,4,7,6] => {{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,5,7,1,3,4,6] => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,5,1,3,4,6,7] => {{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,6,3,7,4,5] => {{1,2},{3,4,6},{5,7}}
 => [2,1,4,6,7,3,5] => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,1,6,7,3,4,5] => {{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
 => [2,4,5,6,7,1,3] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,1,7,3,4,5,6] => {{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => {{1,2},{3},{4},{5},{6},{7}}
 => [2,1,3,4,5,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [3,4,5,1,6,2,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,1,2,7] => {{1,3,5},{2,4,6},{7}}
 => [3,4,5,6,1,2,7] => ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [3,4,5,1,2,7,6] => {{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,5,7,1,2,6] => {{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6,7] => {{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [3,4,1,6,2,5,7] => {{1,3},{2,4,5,6},{7}}
 => [3,4,1,5,6,2,7] => ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,4,6,7,1,2,5] => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5,7] => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => {{1,3},{2,4},{5},{6},{7}}
 => [3,4,1,2,5,6,7] => ? = 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,5,6,2,4,7] => {{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [3,5,1,2,6,4,7] => {{1,3},{2,4,5,6},{7}}
 => [3,4,1,5,6,2,7] => ? = 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [3,5,1,6,2,4,7] => {{1,3},{2,5},{4,6},{7}}
 => [3,5,1,6,2,4,7] => ? = 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [3,5,6,1,7,2,4] => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,5,6,7,1,2,4] => {{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => ? = 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [3,5,6,1,2,4,7] => {{1,3,4,6},{2,5},{7}}
 => [3,5,4,6,2,1,7] => ? = 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,1,2,4,7,6] => {{1,3},{2,4,5},{6,7}}
 => [3,4,1,5,2,7,6] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,7,1,2,4,6] => {{1,3,4,6,7},{2,5}}
 => [3,5,4,6,2,7,1] => ? = 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,5,1,2,4,6,7] => {{1,3},{2,4,5},{6},{7}}
 => [3,4,1,5,2,6,7] => ? = 1

Description

The number of pure excedances of a permutation. A pure excedance of a permutation

$\pi$ is a position

$i < \pi_i$ such that there is no

$j < i$ with

$i\leq \pi_j < \pi_i$ .

The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001737The number of descents of type 2 in a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001060The distinguishing index of a graph. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St001545The second Elser number of a connected graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000862The number of parts of the shifted shape of a permutation. St000007The number of saliances of the permutation. St001597The Frobenius rank of a skew partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001741The largest integer such that all patterns of this size are contained in the permutation. St000542The number of left-to-right-minima of a permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000028The number of stack-sorts needed to sort a permutation. St001624The breadth of a lattice.