- FindStat

Your data matches 199 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001181

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.

Matching statistic: St000296

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,0,0,1,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,0,1,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 100 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 100 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 10 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 101010 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 11010 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 11010 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 100110 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 10100 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 10100 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 10010 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 10010 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1000 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 1000 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 100 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 100 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 100 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 111010 => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 110110 => 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [3,3,2]
 => 110100 => 0
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1

Description

The length of the symmetric border of a binary word. The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix. The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].

Matching statistic: St000629
(load all 4 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,0,0,1,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,0,1,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 100 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 100 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 10 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 101010 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 11010 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 11010 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 100110 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 10100 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 10100 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 10010 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 10010 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1000 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 1000 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 100 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 100 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 100 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 111010 => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 110110 => 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [3,3,2]
 => 110100 => 0
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1

Description

The defect of a binary word. The defect of a finite word

$w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in

$w$ . The maximum possible number of palindromic factors in a word

$w$ is

$|w|+1$ .

Matching statistic: St000687

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

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

Description

The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression,

$I$ is the direct sum of all injective non-projective indecomposable modules and

$P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].

Matching statistic: St001107

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

Matching statistic: St001292

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

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

Description

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here

$A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].

Matching statistic: St001371

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,0,0,1,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,0,1,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 100 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 100 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 10 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 101010 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 11010 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 11010 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 100110 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 10100 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 10100 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 10010 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 10010 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1000 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 1000 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 100 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 100 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 100 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 10 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 111010 => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 110110 => 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [3,3,2]
 => 110100 => 0
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0]
 => [6]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St000326

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,0,1,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 100 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 100 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 11010 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 11010 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 100110 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 10100 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 10100 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1100 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1100 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1100 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1000 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 1000 => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 100 => 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 100 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 100 => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 111010 => 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 110110 => 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [3,3,2]
 => 110100 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [5]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [6]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [5]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => [6]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1]
 => []
 =>  => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0]
 => [6]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
 => [5]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0]
 => [4]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0]
 => [3]
 => []
 =>  => ? = 1 + 1

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000781
(load all 9 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

Matching statistic: St001256

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

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

Description

Number of simple reflexive modules that are 2-stable reflexive. See Definition 3.1. in the reference for the definition of 2-stable reflexive.

The following 189 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001722The number of minimal chains with small intervals between a binary word and the top element. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000264The girth of a graph, which is not a tree. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000741The Colin de Verdière graph invariant. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001330The hat guessing number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000666The number of right tethers of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001208The 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

$K[x]/(x^n)$ . St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001545The second Elser number of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St000944The 3-degree of an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St000862The number of parts of the shifted shape of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001875The number of simple modules with projective dimension at most 1. St000181The number of connected components of the Hasse diagram for the poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000068The number of minimal elements in a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St001820The size of the image of the pop stack sorting operator. St000007The number of saliances of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000834The number of right outer peaks of a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000842The breadth of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000383The last part of an integer composition.