Your data matches 106 different statistics following compositions of up to 3 maps.
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St000480: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 2
[5,2,1]
=> 2
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
St000481: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 1
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 1
[5,3]
=> 1
[5,2,1]
=> 2
Description
The number of upper covers of a partition in dominance order.
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St000837: Permutations ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[4]
=> []
=> []
=> [] => ? = 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[5]
=> []
=> []
=> [] => ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[6]
=> []
=> []
=> [] => ? = 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[7]
=> []
=> []
=> [] => ? ∊ {1,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,2}
[8]
=> []
=> []
=> [] => ? ∊ {1,1,2}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,1,2}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {1,1,2}
[9]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,2}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,2}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {1,1,1,1,2}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,2}
[10]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,1,1,1,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,3}
[11]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[12]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
Description
The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St000836
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000836: Permutations ⟶ ℤResult quality: 78% values known / values provided: 78%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[4]
=> []
=> []
=> [] => ? = 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[5]
=> []
=> []
=> [] => ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[6]
=> []
=> []
=> [] => ? = 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[7]
=> []
=> []
=> [] => ? ∊ {1,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,2}
[8]
=> []
=> []
=> [] => ? ∊ {1,1,2}
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,1,2}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,1,2}
[9]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,2}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,1,1,1,2}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {1,1,1,1,2}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,1,1,1,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {1,1,1,1,2}
[10]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,1,1,1,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {1,1,1,1,1,1,1,3}
[11]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2}
[12]
=> []
=> []
=> [] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00122: Dyck paths Elizalde-Deutsch bijectionDyck paths
St001503: Dyck paths ⟶ ℤResult quality: 75% values known / values provided: 77%distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[8]
=> []
=> []
=> []
=> ? ∊ {0,1}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1}
[9]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1}
[10]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[11]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[12]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[5,5,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[5,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
Matching statistic: St000568
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00072: Permutations binary search tree: left to rightBinary trees
St000568: Binary trees ⟶ ℤResult quality: 74% values known / values provided: 74%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 3 = 2 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 2 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> 2 = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> 3 = 2 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> ? ∊ {1,2,2} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,2,2} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> 3 = 2 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [[.,[.,[.,.]]],[[.,.],.]]
=> 3 = 2 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ? ∊ {1,2,2} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ? ∊ {1,1,2,2,2,2} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> ? ∊ {1,1,2,2,2,2} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,1,2,2,2,2} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? ∊ {1,1,2,2,2,2} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? ∊ {1,1,2,2,2,2} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ? ∊ {1,1,2,2,2,2} + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],[.,.]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,2,7,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [.,[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [.,[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3} + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10,12] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,1,2,3,4,5,6,7,8,11] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,9,2,3,4,5,6,7,10] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,1,2,3,4,5,6,8,10] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,1,2,8,3,4,5,6,9] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,1,2,3,4,5,9] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,1,2,3,4,6,7,9] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,1,7,2,3,4,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,1,2,3,5,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,2,3,9,4,5,6,7] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,10,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,6,9,2,3,4,5,7,8] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,8,10,2,3,4,5,6,7,9] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,2,3,4,5,6,7,8,9] => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,11,1,2,3,4,5,6,7,8,9,12] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,10,2,3,4,5,6,7,8,11] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,10,1,2,3,4,5,6,7,9,11] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,9,3,4,5,6,7,10] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,9,1,2,3,4,5,6,10] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,9,1,2,3,4,5,7,8,10] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,8,4,5,6,9] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [6,7,1,8,2,3,4,5,9] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,8,2,3,4,5,7,9] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
Description
The hook number of a binary tree. A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00122: Dyck paths Elizalde-Deutsch bijectionDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
St001031: Dyck paths ⟶ ℤResult quality: 73% values known / values provided: 73%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,2}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,2,2,2,2,2}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,2,2,2,2,2}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,2,2,2,2,2}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,2,2,2,2,2}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,2,2,2,2}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ?
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ?
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000331
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000331: Dyck paths ⟶ ℤResult quality: 73% values known / values provided: 73%distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2}
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,2,2,2,2,2,2}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,2,2,2,2,2}
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
Description
The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St001509
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001509: Dyck paths ⟶ ℤResult quality: 73% values known / values provided: 73%distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2}
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,2,2,2,2,2,2}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,2,2,2,2,2,2}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,2,2,2,2,2}
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,3,3}
Description
The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$. This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly below the diagonal and relative to the trivial lower boundary.
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000994: Permutations ⟶ ℤResult quality: 69% values known / values provided: 69%distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> []
=> [1] => 0
[2]
=> []
=> []
=> [1] => 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[3]
=> []
=> []
=> [1] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4]
=> []
=> []
=> [1] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[5]
=> []
=> []
=> [1] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[6]
=> []
=> []
=> [1] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
[7]
=> []
=> []
=> [1] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,1}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,1}
[8]
=> []
=> []
=> [1] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,2,2,2}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {1,2,2,2}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,2,2,2}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {1,2,2,2}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,2,2,2,2,2,2}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {1,2,2,2,2,2,2}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,2,2,2,2,2,2}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? ∊ {1,2,2,2,2,2,2}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {1,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? ∊ {1,2,2,2,2,2,2}
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3}
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,12,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[6,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
Description
The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$. Clearly, every cycle of $\pi$ contains as many peaks as valleys.
The following 96 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000201The number of leaf nodes in a binary tree. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001665The number of pure excedances of a permutation. St000092The number of outer peaks of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000353The number of inner valleys of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000640The rank of the largest boolean interval in a poset. St001469The holeyness of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001737The number of descents of type 2 in a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000306The bounce count of a Dyck path. St000259The diameter of a connected graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000470The number of runs in a permutation. St001928The number of non-overlapping descents in a permutation. St000862The number of parts of the shifted shape of a permutation. St000023The number of inner peaks of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000099The number of valleys of a permutation, including the boundary. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000897The number of different multiplicities of parts of an integer partition. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001490The number of connected components of a skew partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000021The number of descents of a permutation. St000354The number of recoils of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St000325The width of the tree associated to a permutation. St001471The magnitude of a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001568The smallest positive integer that does not appear twice in the partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000834The number of right outer peaks of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001729The number of visible descents of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000260The radius of a connected graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001394The genus of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000256The number of parts from which one can substract 2 and still get an integer partition. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. 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 $eA$. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001960The number of descents of a permutation minus one if its first entry is not one. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St000840The number of closers smaller than the largest opener in a perfect matching. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001728The number of invisible descents of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000454The largest eigenvalue of a graph if it is integral. St001948The number of augmented double ascents of a permutation. St001114The number of odd descents of a permutation. St001642The Prague dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001896The number of right descents of a signed permutations. 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 $eA$.