Processing math: 100%

Your data matches 222 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001137
(load all 63 compositions to match this statistic)
Mapping
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0
[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St000205
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 70%●distinct values known / distinct values provided: 50%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,1] => [1,1]
 => [1]
 => 0
[1,1,0,0]
 => [2] => [2]
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [1]
 => 0
[1,1,0,1,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,1,1,0,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [4,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given λ count how many ''integer partitions'' w (weight) there are, such that Pλ,w is non-integral, i.e., w such that the Gelfand-Tsetlin polytope Pλ,w has at least one non-integral vertex.
Matching statistic: St000206
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 70%●distinct values known / distinct values provided: 50%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,1] => [1,1]
 => [1]
 => 0
[1,1,0,0]
 => [2] => [2]
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [1]
 => 0
[1,1,0,1,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,1,1,0,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [4,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given λ count how many ''integer compositions'' w (weight) there are, such that Pλ,w is non-integral, i.e., w such that the Gelfand-Tsetlin polytope Pλ,w has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000749
(load all 4 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,1] => [1,1]
 => [1]
 => 0
[1,1,0,0]
 => [2] => [2]
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [1]
 => 0
[1,1,0,1,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,1,1,0,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [4,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
Description
The 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. For example, restricting S(6,3) to S8 yields S(5,3)⊕S(6,2) of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to S7 yields S(4,3)⊕2S(5,2)⊕S(6,1) of degrees 14, 14 and 6. However, restricting to S6 yields S(3,3)⊕3S(4,2)⊕3S(5,1)⊕S6 of degrees 5,9,5 and 1. Therefore, the statistic on the partition (6,3) gives 3. This is related to 2-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001175
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,1] => [1,1]
 => [1]
 => 0
[1,1,0,0]
 => [2] => [2]
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [1]
 => 0
[1,1,0,1,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,1,1,0,0,0]
 => [3] => [3]
 => []
 => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => [1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [4,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000661
(load all 5 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 0
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 0
[1,1,0,0]
 => [2] => [1] => [1,0]
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [3] => [1] => [1,0]
 => ? ∊ {0,0}
[1,1,1,0,0,0]
 => [3] => [1] => [1,0]
 => ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2] => [1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St000980
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 69%●distinct values known / distinct values provided: 25%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 0
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 0
[1,1,0,0]
 => [2] => [1] => [1,0]
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [3] => [1] => [1,0]
 => ? ∊ {0,1}
[1,1,1,0,0,0]
 => [3] => [1] => [1,0]
 => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2] => [1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path 111011010000 has three peaks in positions 03,15,26. The boxes below 03 are 01,02,12, the boxes below 15 are 12,13,14,23,24,34, and the boxes below 26 are 23,24,25,34,35,45. We thus obtain the four boxes in positions 12,23,24,34 that are below at least two peaks.
Matching statistic: St001141
(load all 8 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 0
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 0
[1,1,0,0]
 => [2] => [1] => [1,0]
 => ? = 0
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [3] => [1] => [1,0]
 => ? ∊ {0,0}
[1,1,1,0,0,0]
 => [3] => [1] => [1,0]
 => ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => [2] => [1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4] => [1] => [1,0]
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,0,1,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1] => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,0,1,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6] => [1] => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.
Matching statistic: St001604
(load all 11 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,0}
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? ∊ {0,0}
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,1}
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => [2,2,1]
 => [2,1]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [2,2,1]
 => [2,1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [2,2,1]
 => [2,1]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [3,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [4,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [3,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [3,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => [4,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => [4,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [4,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,6,4,5,2] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => [3,2,1]
 => [2,1]
 => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => [2,2,1,1]
 => [2,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,3,5,2,6] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [4,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,3,6,5,2] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,5,6,3] => [4,2]
 => [2]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,1,5,6] => [4,1,1]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001714
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 52%●distinct values known / distinct values provided: 25%
Values
[1,0]
 => [1] => [1,0]
 => []
 => ? = 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => []
 => ? = 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,1}
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? ∊ {0,1}
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 0
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1}
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1}
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic 0 are wide partitions.
The following 212 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001586The number of odd parts smaller than the largest even part in an integer partition. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001172The number of 1-rises at odd height of a Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000306The bounce count of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000296The length of the symmetric border of a binary word. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000629The defect of a binary word. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001524The degree of symmetry of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000117The number of centered tunnels of a Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000546The number of global descents of a permutation. St000478Another weight of a partition according to Alladi. St000214The number of adjacencies of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St001176The size of a partition minus its first part. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. 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. St001877Number of indecomposable injective modules with projective dimension 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St000031The number of cycles in the cycle decomposition of a permutation. St000664The number of right ropes of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001875The number of simple modules with projective dimension at most 1. St000563The number of overlapping pairs of blocks of a set partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001330The hat guessing number of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001964The interval resolution global dimension of a poset. St000650The number of 3-rises of a permutation. St000654The first descent of a permutation. St000366The number of double descents of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001549The number of restricted non-inversions between exceedances. St001811The Castelnuovo-Mumford regularity of a permutation. St000990The first ascent of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001399The distinguishing number of a poset. St000534The number of 2-rises of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St000633The size of the automorphism group of a poset. St001162The minimum jump of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St000022The number of fixed points of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St000731The number of double exceedences of a permutation. St000989The number of final rises of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000527The width of the poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001846The number of elements which do not have a complement in the lattice. St001096The size of the overlap set of a permutation. St000842The breadth of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000352The Elizalde-Pak rank of a permutation. St000058The order of a permutation. St001820The size of the image of the pop stack sorting operator. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001868The number of alignments of type NE of a signed permutation. St000648The number of 2-excedences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St001513The number of nested exceedences of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000768The number of peaks in an integer composition. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000741The Colin de Verdière graph invariant. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St000920The logarithmic height of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001394The genus of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001948The number of augmented double ascents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000805The number of peaks of the associated bargraph. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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]/(xn). St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001737The number of descents of type 2 in a permutation. St000542The number of left-to-right-minima of a permutation. St001481The minimal height of a peak of a Dyck path. St000764The number of strong records in an integer composition. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000883The number of longest increasing subsequences of a permutation.