Your data matches 66 different statistics following compositions of up to 3 maps.
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St000003: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 6
[2,2,1]
=> 5
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 5
[4,2]
=> 9
[4,1,1]
=> 10
[3,3]
=> 5
[3,2,1]
=> 16
[3,1,1,1]
=> 10
[2,2,2]
=> 5
[2,2,1,1]
=> 9
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 6
[5,2]
=> 14
[5,1,1]
=> 15
[4,3]
=> 14
[4,2,1]
=> 35
[4,1,1,1]
=> 20
[3,3,1]
=> 21
[3,2,2]
=> 21
[3,2,1,1]
=> 35
[3,1,1,1,1]
=> 15
[2,2,2,1]
=> 14
[2,2,1,1,1]
=> 14
[2,1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1]
=> 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Mp00179: Integer partitions to skew partitionSkew partitions
St001595: Skew partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> 1
[2]
=> [[2],[]]
=> 1
[1,1]
=> [[1,1],[]]
=> 1
[3]
=> [[3],[]]
=> 1
[2,1]
=> [[2,1],[]]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> 1
[4]
=> [[4],[]]
=> 1
[3,1]
=> [[3,1],[]]
=> 3
[2,2]
=> [[2,2],[]]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> 3
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[5]
=> [[5],[]]
=> 1
[4,1]
=> [[4,1],[]]
=> 4
[3,2]
=> [[3,2],[]]
=> 5
[3,1,1]
=> [[3,1,1],[]]
=> 6
[2,2,1]
=> [[2,2,1],[]]
=> 5
[2,1,1,1]
=> [[2,1,1,1],[]]
=> 4
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[6]
=> [[6],[]]
=> 1
[5,1]
=> [[5,1],[]]
=> 5
[4,2]
=> [[4,2],[]]
=> 9
[4,1,1]
=> [[4,1,1],[]]
=> 10
[3,3]
=> [[3,3],[]]
=> 5
[3,2,1]
=> [[3,2,1],[]]
=> 16
[3,1,1,1]
=> [[3,1,1,1],[]]
=> 10
[2,2,2]
=> [[2,2,2],[]]
=> 5
[2,2,1,1]
=> [[2,2,1,1],[]]
=> 9
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 5
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 1
[7]
=> [[7],[]]
=> 1
[6,1]
=> [[6,1],[]]
=> 6
[5,2]
=> [[5,2],[]]
=> 14
[5,1,1]
=> [[5,1,1],[]]
=> 15
[4,3]
=> [[4,3],[]]
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> 35
[4,1,1,1]
=> [[4,1,1,1],[]]
=> 20
[3,3,1]
=> [[3,3,1],[]]
=> 21
[3,2,2]
=> [[3,2,2],[]]
=> 21
[3,2,1,1]
=> [[3,2,1,1],[]]
=> 35
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> 15
[2,2,2,1]
=> [[2,2,2,1],[]]
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 14
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> 6
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> 1
Description
The number of standard Young tableaux of the skew partition.
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
St000100: Posets ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 6
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 16
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 10
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 9
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 6
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 14
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 15
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 20
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 21
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 35
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 15
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 14
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 6
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The number of linear extensions of a poset.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000001: Permutations ⟶ ℤResult quality: 95% values known / values provided: 95%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 6
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 9
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 10
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 5
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 16
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 10
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 9
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 5
[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
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 14
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 15
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 14
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 35
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 21
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 21
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 35
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 15
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 14
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 14
[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] => 6
[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
Description
The number of reduced words for a permutation. This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St001877
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
Mp00195: Posets order idealsLattices
St001877: Lattices ⟶ ℤResult quality: 21% values known / values provided: 32%distinct values known / distinct values provided: 21%
Values
[1]
=> [[1],[]]
=> ([],1)
=> ([(0,1)],2)
=> ? = 1 - 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 4 - 1
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5 - 1
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 6 - 1
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5 - 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 4 - 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 5 - 1
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 9 - 1
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 10 - 1
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 5 - 1
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ? = 16 - 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 10 - 1
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 5 - 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 9 - 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 5 - 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? = 6 - 1
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? = 14 - 1
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? = 15 - 1
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? = 14 - 1
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? = 35 - 1
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
=> ? = 20 - 1
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? = 21 - 1
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? = 21 - 1
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? = 35 - 1
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? = 15 - 1
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? = 14 - 1
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? = 14 - 1
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? = 6 - 1
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001768
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001768: Signed permutations ⟶ ℤResult quality: 30% values known / values provided: 30%distinct values known / distinct values provided: 43%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,1,2] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 6
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [6,2,1,3,4,5] => ? = 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 9
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,2,3,1,4] => ? = 10
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 5
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 16
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => ? = 10
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,5,1,2] => ? = 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ? = 9
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => ? = 5
[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] => [2,3,4,5,6,7,1] => ? = 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [7,2,1,3,4,5,6] => ? = 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,2,4,5] => ? = 14
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [6,2,3,1,4,5] => ? = 15
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 14
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,3,2,1,4] => ? = 35
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => ? = 20
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => ? = 21
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 21
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 35
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [4,2,3,5,6,1] => ? = 15
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ? = 14
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,4,2,5,6,1] => ? = 14
[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] => [3,2,4,5,6,7,1] => ? = 6
[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] => [2,3,4,5,6,7,8,1] => ? = 1
Description
The number of reduced words of a signed permutation. This is the number of ways to write a permutation as a minimal length product of simple reflections.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00082: Standard tableaux to Gelfand-Tsetlin patternGelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 21% values known / values provided: 23%distinct values known / distinct values provided: 21%
Values
[1]
=> [[1]]
=> [[1]]
=> ? = 1
[2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 1
[2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
[2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1
[4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 4
[3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
[6]
=> [[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 5
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 16
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 10
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 5
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 9
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 14
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 15
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 14
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[4,2,1,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 35
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 20
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 21
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 21
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[3,2,1,1,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 35
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 15
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 14
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 14
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 6
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1
Description
The order of promotion on a Gelfand-Tsetlin pattern.
Mp00313: Integer partitions Glaisher-Franklin inverseInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00093: Dyck paths to binary wordBinary words
St001722: Binary words ⟶ ℤResult quality: 14% values known / values provided: 16%distinct values known / distinct values provided: 14%
Values
[1]
=> [1]
=> [1,0]
=> 10 => 1
[2]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1
[1,1]
=> [2]
=> [1,0,1,0]
=> 1010 => 1
[3]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 1
[2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2
[1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1
[3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 3
[2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 2
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3
[1,1,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1
[5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1
[4,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 4
[3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 5
[3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 6
[2,2,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 5
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 4
[1,1,1,1,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 1
[6]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => ? = 1
[5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 5
[4,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 9
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 10
[3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 5
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 16
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 10
[2,2,2]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 5
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 9
[2,1,1,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 5
[1,1,1,1,1,1]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 1
[7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? = 1
[6,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 6
[5,2]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? = 14
[5,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? = 15
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => ? = 14
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? = 35
[4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => ? = 20
[3,3,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? = 21
[3,2,2]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 21
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => ? = 35
[3,1,1,1,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 15
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 14
[2,2,1,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => ? = 14
[2,1,1,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => ? = 6
[1,1,1,1,1,1,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000365
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000365: Permutations ⟶ ℤResult quality: 14% values known / values provided: 16%distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ? = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 6 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 5 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 4 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 9 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => ? = 10 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 5 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 16 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => ? = 10 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 5 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 9 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => ? = 5 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => ? = 14 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? = 15 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 14 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => ? = 35 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
=> [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? = 20 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 21 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 21 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 35 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
=> [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? = 15 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 14 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 14 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1 - 1
Description
The number of double ascents of a permutation. A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St001487
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00233: Dyck paths skew partitionSkew partitions
St001487: Skew partitions ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? = 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? = 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? = 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> ? = 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? = 6
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> ? = 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2],[]]
=> ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[7],[]]
=> ? = 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[6,6],[4]]
=> ? = 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 9
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[5,5,5],[3,3]]
=> ? = 10
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 5
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> ? = 16
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,2]]
=> ? = 10
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 5
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> ? = 9
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1,1,1]]
=> ? = 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2,2],[]]
=> ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[8],[]]
=> ? = 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[7,7],[5]]
=> ? = 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[6,6],[3]]
=> ? = 14
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[6,6,6],[4,4]]
=> ? = 15
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[5,5],[1]]
=> ? = 14
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[5,5,5],[3,2]]
=> ? = 35
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[5,5,5,5],[3,3,3]]
=> ? = 20
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> ? = 21
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 21
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,1]]
=> ? = 35
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[4,4,4,4,4],[2,2,2,2]]
=> ? = 15
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> ? = 14
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1,1]]
=> ? = 14
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3,3],[1,1,1,1,1]]
=> ? = 6
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2,2,2],[]]
=> ? = 1
Description
The number of inner corners of a skew partition.
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001488The number of corners of a skew partition. St000068The number of minimal elements in a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000234The number of global ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001344The neighbouring number of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001712The number of natural descents of a standard Young tableau. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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 $eA$. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001928The number of non-overlapping descents in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001424The number of distinct squares in a binary word.