- FindStat

Your data matches 103 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000237
(load all 13 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St000010

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000382
(load all 3 compositions to match this statistic)

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1,0]
 => [1,0]
 => [1] => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1] => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 2 = 1 + 1
[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]
 => [3,1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1 = 0 + 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]
 => [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => 3 = 2 + 1
[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]
 => [3,1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,1,1,1,1,3] => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,2,1,1,1,1,3] => ? = 0 + 1
[]
 => ?
 => ?
 => ? => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,9] => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [11] => ? = 10 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,9] => ? = 1 + 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,1,1,5] => ? = 1 + 1
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? => ? = 0 + 1
[1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => ?
 => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [7,1,1,1,1,1] => ? = 6 + 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ?
 => ?
 => ? => ? = 0 + 1
[1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ?
 => ? => ? = 1 + 1
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => ?
 => ? => ? = 1 + 1
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => ? => ? = 1 + 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ?
 => ?
 => ? => ? = 0 + 1
[1,1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ?
 => ?
 => ? => ? = 0 + 1
[1,1,1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ?
 => ?
 => ? => ? = 0 + 1
[1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => ?
 => ? => ? = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
 => ?
 => ? => ? = 0 + 1

Description

The first part of an integer composition.

Matching statistic: St000288
(load all 2 compositions to match this statistic)

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 95%●distinct values known / distinct values provided: 82%

Values

[1,0]
 => [1,0]
 => [1] => 1 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [2] => 10 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => 11 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => 100 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => 101 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,2] => 110 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 111 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [3] => 100 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => 1000 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => 1000 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => 1000 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 10001 => 2 = 1 + 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 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => 10000 => 1 = 0 + 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 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => 10000 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => 10001 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => 10000 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => 10000 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 2 = 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] => 11001 => 3 = 2 + 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 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 11100 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 11000 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 11001 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 11000 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 11000 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => 100000001 => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [8,1] => 100000001 => ? = 1 + 1
[]
 => []
 => [] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => 100000001 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => 100000001 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => 1000000001 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [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 => ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1] => 10000000001 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1] => 11111111111 => ? = 10 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,2,2,1] => 1101010101 => ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,10] => 11000000000 => ? = 1 + 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => 1000000001 => ? = 1 + 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1 + 1
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,2,2,2,1] => 110101010101 => ? = 6 + 1
[1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 1 + 1
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,9] => 1100000000 => ? = 1 + 1
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,11] => 110000000000 => ? = 1 + 1
[1,1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => ? => ? => ? = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? => ? => ? = 0 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000097

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [2] => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(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]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 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] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [11] => ([],11)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [11] => ([],11)
 => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[]
 => []
 => [] => ?
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 6 + 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 10 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,9] => ([(8,9)],10)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 8 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,10] => ([(9,10)],11)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9] => ([(8,9)],10)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [11] => ([],11)
 => ? = 0 + 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,9] => ([(8,9)],10)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St001581

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%

Values

[1,0]
 => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [2] => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(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]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 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] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [11] => ([],11)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [11] => ([],11)
 => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[]
 => []
 => [] => ?
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [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] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 8 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 6 + 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [9] => ([],9)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [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] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 10 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,9] => ([(8,9)],10)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [10] => ([],10)
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 6 + 1

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

Matching statistic: St000011
(load all 28 compositions to match this statistic)

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[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] => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 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] => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 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] => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 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] => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 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] => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 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] => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 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] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,8,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,4,1,6,8,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [2,3,4,6,5,1,8,7] => ?
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,4,7,8,6] => ?
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,1,4,5,6,3,8,7] => [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,4,5,6,8,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,3,6,7,8,5] => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,3,6,8,7,5] => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,1,4,6,5,7,8,3] => ?
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,1,4,6,5,3,8,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,1,4,6,5,8,7,3] => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,1,4,7,5,6,8,3] => ?
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4,6,8,7] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [2,4,3,5,6,1,8,7] => [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [2,4,3,1,6,7,8,5] => [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => [2,5,3,4,6,1,8,7] => ?
 => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [2,7,6,4,5,3,1,8] => ?
 => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [2,8,6,4,5,7,3,1] => ?
 => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,8,6,4,5,3,7,1] => ?
 => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,7,2,8] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,3,4,8,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,2,5,7,6,4,8] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,2,5,8,7,6,4] => [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,2,5,8,6,7,4] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [1,3,5,4,2,7,6,8] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,3,5,4,7,6,2,8] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,3,6,5,4,7,2,8] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,3,8,5,4,7,6,2] => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,3,6,4,5,7,2,8] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,3,6,4,5,2,7,8] => ?
 => ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,3,7,4,6,5,2,8] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,3,8,4,5,6,7,2] => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,7,5,6,4,2] => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,6,3,7,8] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 4 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,6,5,4,3,7,8] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [1,4,3,5,7,6,2,8] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? = 2 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,7,4,3,6,5,2,8] => ?
 => ? = 2 + 1
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,8,4,3,7,6,5,2] => ?
 => ? = 1 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,7,6,4,5,3,8,2] => ?
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => [3,2,4,7,6,5,1,8] => ?
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,7,6,8] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [3,2,1,5,4,8,7,6] => ?
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => [3,2,1,5,7,6,4,8] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,5,4,1,7,6,8] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [3,2,5,4,7,6,1,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
 => [3,2,7,4,6,5,1,8] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
 => [4,3,2,5,1,7,6,8] => ?
 => ? = 2 + 1
[1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
 => [8,3,4,2,7,6,5,1] => ?
 => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
 => [4,2,3,5,1,7,6,8] => ?
 => ? = 2 + 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: St000439
(load all 16 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St000925
(load all 5 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 72%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St000678
(load all 27 compositions to match this statistic)

Mapping

Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 73%

Values

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

Description

The number of up steps after the last double rise of a Dyck path.

The following 93 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000675The number of centered multitunnels of a Dyck path. St000214The number of adjacencies of a permutation. St001050The number of terminal closers of a set partition. St000098The chromatic number of a graph. St000546The number of global descents of a permutation. St000069The number of maximal elements of a poset. St000971The smallest closer of a set partition. St000054The first entry of the permutation. St000007The number of saliances of the permutation. St000504The cardinality of the first block of a set partition. St000068The number of minimal elements in a poset. St000502The number of successions of a set partitions. St000234The number of global ascents of a permutation. St000025The number of initial rises of a Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001479The number of bridges of a graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000843The decomposition number of a perfect matching. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000654The first descent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000740The last entry of a permutation. St000738The first entry in the last row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000822The Hadwiger number of the graph. St000990The first ascent of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000883The number of longest increasing subsequences of a permutation. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St000648The number of 2-excedences of a permutation. St001330The hat guessing number of a graph. St000534The number of 2-rises of a permutation. St000924The number of topologically connected components of a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St000080The rank of the poset. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001889The size of the connectivity set of a signed permutation. St000022The number of fixed points of a permutation. St000649The number of 3-excedences of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions.