- FindStat

Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000292

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a binary word.

Matching statistic: St001732
(load all 10 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.

Matching statistic: St001330
(load all 3 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%

Values

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

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St000455

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000765

Mapping

Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1,0,1,0]
 => [[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1] => 1 = 0 + 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => [[1,2,3,4]]
 => [4] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [[1,2,4,6],[3,5]]
 => [3,2,1] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [[1,2,4,5,6],[3]]
 => [3,3] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [[1,2,3,5,6],[4]]
 => [4,2] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [[1,2,3,4,5,6]]
 => [6] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [[1,2,4,6,8],[3,5,7]]
 => [3,2,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [[1,2,4,6,7,8],[3,5]]
 => [3,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [[1,2,4,5,6,8],[3,7]]
 => [3,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [[1,2,4,5,7,8],[3,6]]
 => [3,3,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [[1,2,4,5,6,7,8],[3]]
 => [3,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [[1,2,3,4,6,8],[5,7]]
 => [5,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [[1,2,3,4,7,8],[5,6]]
 => [5,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [[1,2,3,5,6,8],[4,7]]
 => [4,3,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => [[1,2,3,5,7,8],[4,6]]
 => [4,2,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => [[1,2,3,5,6,7,8],[4]]
 => [4,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [[1,2,3,4,5,6,8],[7]]
 => [7,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => [[1,2,3,4,5,7,8],[6]]
 => [6,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => [[1,2,3,4,6,7,8],[5]]
 => [5,3] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,2,3,4,5,6,7,8]]
 => [8] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [[1,2,4,6,8,10],[3,5,7,9]]
 => [3,2,2,2,1] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [[1,2,4,6,8,9,10],[3,5,7]]
 => [3,2,2,3] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [[1,2,4,6,7,8,10],[3,5,9]]
 => [3,2,4,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => [[1,2,4,6,7,9,10],[3,5,8]]
 => [3,2,3,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [[1,2,4,6,7,8,9,10],[3,5]]
 => [3,2,5] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [[1,2,4,5,6,8,10],[3,7,9]]
 => [3,4,2,1] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [[1,2,4,5,6,9,10],[3,7,8]]
 => [3,4,3] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [[1,2,4,5,7,8,10],[3,6,9]]
 => [3,3,3,1] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [[1,2,4,5,7,9,10],[3,6,8]]
 => [3,3,2,2] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [[1,2,4,5,7,8,9,10],[3,6]]
 => [3,3,4] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [[1,2,4,5,6,7,8,10],[3,9]]
 => [3,6,1] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [[1,2,4,5,6,7,9,10],[3,8]]
 => [3,5,2] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [[1,2,4,5,6,8,9,10],[3,7]]
 => [3,4,3] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [[1,2,4,5,6,7,8,9,10],[3]]
 => [3,7] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [[1,2,3,4,6,8,10],[5,7,9]]
 => [5,2,2,1] => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [[1,2,3,4,6,9,10],[5,7,8]]
 => [5,2,3] => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [[1,2,3,4,7,8,10],[5,6,9]]
 => [5,4,1] => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [[1,2,3,4,7,9,10],[5,6,8]]
 => [5,3,2] => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [[1,2,3,4,7,8,9,10],[5,6]]
 => [5,5] => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [[1,2,3,5,6,8,10],[4,7,9]]
 => [4,3,2,1] => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [[1,2,3,5,6,9,10],[4,7,8]]
 => [4,3,3] => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [[1,2,3,5,7,8,10],[4,6,9]]
 => [4,2,3,1] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [[1,2,3,5,7,9,10],[4,6,8]]
 => [4,2,2,2] => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [[1,2,3,5,7,8,9,10],[4,6]]
 => [4,2,4] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [[1,2,3,5,6,7,8,10],[4,9]]
 => [4,5,1] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [[1,2,3,5,6,7,9,10],[4,8]]
 => [4,4,2] => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [[1,2,3,5,6,8,9,10],[4,7]]
 => [4,3,3] => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [[1,2,3,5,6,7,8,9,10],[4]]
 => [4,6] => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => [[1,2,3,4,5,6,8,10],[7,9]]
 => [7,2,1] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [[1,2,3,4,5,6,9,10],[7,8]]
 => [7,3] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => [[1,2,3,4,5,7,8,10],[6,9]]
 => [6,3,1] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => [[1,2,3,4,5,7,9,10],[6,8]]
 => [6,2,2] => ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [[1,2,3,6,7],[4,5,8,9,10]]
 => [[1,2,3,4,5,8,9,10],[6,7]]
 => [6,4] => ? = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => [[1,2,3,4,6,7,8,10],[5,9]]
 => [5,4,1] => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => [[1,2,3,4,6,7,9,10],[5,8]]
 => [5,3,2] => ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [[1,2,3,5,7],[4,6,8,9,10]]
 => [[1,2,3,4,6,8,9,10],[5,7]]
 => [5,2,3] => ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => [[1,2,3,4,6,7,8,9,10],[5]]
 => [5,5] => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => [[1,2,3,4,5,6,7,8,10],[9]]
 => [9,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [[1,2,3,4,8],[5,6,7,9,10]]
 => [[1,2,3,4,5,6,7,9,10],[8]]
 => [8,2] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [[1,2,3,4,7],[5,6,8,9,10]]
 => [[1,2,3,4,5,6,8,9,10],[7]]
 => [7,3] => ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [[1,2,3,4,6],[5,7,8,9,10]]
 => [[1,2,3,4,5,7,8,9,10],[6]]
 => [6,4] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [10] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => [[1,2,4,6,8,10,12],[3,5,7,9,11]]
 => [3,2,2,2,2,1] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => [[1,2,4,6,8,10,11,12],[3,5,7,9]]
 => [3,2,2,2,3] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => [[1,2,4,6,8,9,10,12],[3,5,7,11]]
 => [3,2,2,4,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,7,8,10],[2,4,6,9,11,12]]
 => [[1,2,4,6,8,9,11,12],[3,5,7,10]]
 => [3,2,2,3,2] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => [[1,2,4,6,8,9,10,11,12],[3,5,7]]
 => [3,2,2,5] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => [[1,2,4,6,7,8,10,12],[3,5,9,11]]
 => [3,2,4,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => [[1,2,4,6,7,8,11,12],[3,5,9,10]]
 => [3,2,4,3] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,3,5,6,8,11],[2,4,7,9,10,12]]
 => [[1,2,4,6,7,9,10,12],[3,5,8,11]]
 => [3,2,3,3,1] => ? = 1 + 1

Description

The number of weak records in an integer composition. A weak record is an element

$a_i$ such that

$a_i \geq a_j$ for all

$j < i$ .

Matching statistic: St001208

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St001737

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of descents of type 2 in a permutation. A position

$i\in[1,n-1]$ is a descent of type 2 of a permutation

$\pi$ of

$n$ letters, if it is a descent and if

$\pi(j) < \pi(i)$ for all

$j < i$ .