Processing math: 100%

Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St001564
St001564: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 3
[3]
=> 1
[2,1]
=> 6
[1,1,1]
=> 10
[4]
=> 1
[3,1]
=> 6
[2,2]
=> 3
[2,1,1]
=> 30
[1,1,1,1]
=> 35
[5]
=> 1
[4,1]
=> 6
[3,2]
=> 6
[3,1,1]
=> 30
[2,2,1]
=> 30
[2,1,1,1]
=> 140
[1,1,1,1,1]
=> 126
[6]
=> 1
[5,1]
=> 6
[4,2]
=> 6
[4,1,1]
=> 30
[3,3]
=> 3
[3,2,1]
=> 60
[3,1,1,1]
=> 140
[2,2,2]
=> 10
[2,2,1,1]
=> 210
[2,1,1,1,1]
=> 630
[1,1,1,1,1,1]
=> 462
[7]
=> 1
[6,1]
=> 6
[5,2]
=> 6
[5,1,1]
=> 30
[4,3]
=> 6
[4,2,1]
=> 60
[4,1,1,1]
=> 140
[3,3,1]
=> 30
[3,2,2]
=> 30
[3,2,1,1]
=> 420
[3,1,1,1,1]
=> 630
[2,2,2,1]
=> 140
[2,2,1,1,1]
=> 1260
[2,1,1,1,1,1]
=> 2772
[1,1,1,1,1,1,1]
=> 1716
Description
The value of the forgotten symmetric functions when all variables set to 1. Let fλ(x) denote the forgotten symmetric functions. Then the statistic associated with λ, where λ has parts, is fλ(1,1,,1) where there are variables substituted by 1.
Matching statistic: St001632
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00065: Permutations permutation posetPosets
St001632: Posets ⟶ ℤResult quality: 6% values known / values provided: 14%distinct values known / distinct values provided: 6%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> ? = 1
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 3
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? ∊ {6,10}
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ? ∊ {6,10}
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {3,6,30,35}
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {3,6,30,35}
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {3,6,30,35}
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ? ∊ {3,6,30,35}
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {6,6,30,30,126,140}
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {6,6,30,30,126,140}
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {6,6,30,30,126,140}
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {6,6,30,30,126,140}
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ? ∊ {6,6,30,30,126,140}
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> ? ∊ {6,6,30,30,126,140}
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> ? ∊ {3,6,6,10,30,60,140,210,462,630}
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(5,6)],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([],7)
=> ? ∊ {6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
Description
The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset.
Matching statistic: St000725
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000725: Permutations ⟶ ℤResult quality: 12% values known / values provided: 14%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 4 = 3 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {6,10} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {6,10} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,6,30,35} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {1,6,30,35} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 4 = 3 + 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] => ? ∊ {1,6,30,35} + 1
[1,1,1,1]
=> [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] => ? ∊ {1,6,30,35} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,6,6,30,30,126,140} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,6,6,30,30,126,140} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {1,6,6,30,30,126,140} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {1,6,6,30,30,126,140} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {1,6,6,30,30,126,140} + 1
[2,1,1,1]
=> [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] => ? ∊ {1,6,6,30,30,126,140} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,6,6,30,30,126,140} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 1 + 1
[3,1,1,1]
=> [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] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {3,6,6,10,30,60,140,210,462,630} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,2,1,1]
=> [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] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,2,2,1]
=> [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] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000541
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000541: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ? ∊ {6,10}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? ∊ {6,10}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => ? ∊ {1,3,6,30,35}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ? ∊ {1,3,6,30,35}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => ? ∊ {1,3,6,30,35}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? ∊ {1,3,6,30,35}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,30,35}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {1,6,6,30,30,126,140}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => ? ∊ {1,6,6,30,30,126,140}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {1,6,6,30,30,126,140}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {1,6,6,30,30,126,140}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {1,6,6,30,30,126,140}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => ? ∊ {1,6,6,30,30,126,140}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,30,30,126,140}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [14,13,12,11,10,9,8,7,6,5,4,3,2,1,16,15] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [12,11,10,9,6,5,8,7,4,3,2,1,14,13] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [10,9,6,5,4,3,8,7,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [10,9,4,3,8,7,6,5,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,12,11,8,7,6,5,10,9,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,12,11,6,5,10,9,8,7,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,14,13,12,11,8,7,10,9,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,16,15,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation π of length n, this is the number of indices 2jn such that for all 1i<j, the pair (i,j) is an inversion of π.
Matching statistic: St001207
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00067: Permutations Foata bijectionPermutations
St001207: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,1,4,3] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,1,5,4] => ? ∊ {6,10}
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,4,1,5,2] => ? ∊ {6,10}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,3,4,1,6,5] => ? ∊ {1,3,6,30,35}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,2,4] => ? ∊ {1,3,6,30,35}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => ? ∊ {1,3,6,30,35}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => ? ∊ {1,3,6,30,35}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,4,1,5,6,2] => ? ∊ {1,3,6,30,35}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,3,4,5,1,7,6] => ? ∊ {1,6,6,30,30,126,140}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,3,4,6,2,5] => ? ∊ {1,6,6,30,30,126,140}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,3,5,4] => ? ∊ {1,6,6,30,30,126,140}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? ∊ {1,6,6,30,30,126,140}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => ? ∊ {1,6,6,30,30,126,140}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,4,2,6,5,3] => ? ∊ {1,6,6,30,30,126,140}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,4,1,5,6,7,2] => ? ∊ {1,6,6,30,30,126,140}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [1,3,4,5,7,2,6] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,1,6,4,3,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,2,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,6,1,5,2,4] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,5,1,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,4,1,6,2,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,4,2,5,7,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[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] => [3,4,1,5,6,7,8,2] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,9,8] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [1,3,4,5,6,8,2,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,1,4,7,5,3,6] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,4,1,7,2,6] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,1,4,6,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,3,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,6,3,2,4] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,3,5,4] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,5,2,6,4] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,5,2,6,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [4,6,1,5,7,2,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[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] => [1,4,2,5,6,8,7,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[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] => [3,4,1,5,6,7,8,9,2] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
Description
The 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]/(xn).
Matching statistic: St001778
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St001778: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ? ∊ {6,10}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? ∊ {6,10}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => ? ∊ {1,3,6,30,35}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ? ∊ {1,3,6,30,35}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => ? ∊ {1,3,6,30,35}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? ∊ {1,3,6,30,35}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,30,35}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {1,6,6,30,30,126,140}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => ? ∊ {1,6,6,30,30,126,140}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {1,6,6,30,30,126,140}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {1,6,6,30,30,126,140}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {1,6,6,30,30,126,140}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => ? ∊ {1,6,6,30,30,126,140}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,30,30,126,140}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [14,13,12,11,10,9,8,7,6,5,4,3,2,1,16,15] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [12,11,10,9,6,5,8,7,4,3,2,1,14,13] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [10,9,6,5,4,3,8,7,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [10,9,4,3,8,7,6,5,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,12,11,8,7,6,5,10,9,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,12,11,6,5,10,9,8,7,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,14,13,12,11,8,7,10,9,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,16,15,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
Description
The largest greatest common divisor of an element and its image in a permutation.
Matching statistic: St001816
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
Mp00106: Standard tableaux catabolismStandard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> ? ∊ {6,10}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> ? ∊ {6,10}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,2,3,4,5,6,7,8,10],[9]]
=> ? ∊ {1,3,6,30,35}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> ? ∊ {1,3,6,30,35}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> ? ∊ {1,3,6,30,35}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> ? ∊ {1,3,6,30,35}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2,4,5,6,7,8,9,10],[3]]
=> ? ∊ {1,3,6,30,35}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> [[1,2,3,4,5,6,7,8,9,10,12],[11]]
=> ? ∊ {1,6,6,30,30,126,140}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,2,3,4,6,7,8,10],[5,9]]
=> ? ∊ {1,6,6,30,30,126,140}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> ? ∊ {1,6,6,30,30,126,140}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> ? ∊ {1,6,6,30,30,126,140}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> ? ∊ {1,6,6,30,30,126,140}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2,4,5,6,8,9,10],[3,7]]
=> ? ∊ {1,6,6,30,30,126,140}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> [[1,2,4,5,6,7,8,9,10,11,12],[3]]
=> ? ∊ {1,6,6,30,30,126,140}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,14],[13]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> [[1,2,3,4,5,7,8,9,10,12],[6,11]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,2,3,4,5,7,8,10],[6,9]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,2,3,5,6,7,8,10],[4,9]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,2,3,4,5,6,9,10],[7,8]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2,4,5,6,7,9,10],[3,8]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,2,3,4,7,8,9,10],[5,6]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2,4,5,7,8,9,10],[3,6]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> [[1,2,4,5,6,7,9,10,11,12],[3,8]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [[1,2,4,5,6,7,8,9,10,11,12,13,14],[3]]
=> ? ∊ {1,3,6,6,10,30,60,140,210,462,630}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,16],[15]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> [[1,2,3,4,5,6,8,9,10,11,12,14],[7,13]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,7,11],[5,6,8,9,10,12]]
=> [[1,2,3,4,5,6,8,9,10,12],[7,11]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[1,2,3,5,6,11],[4,7,8,9,10,12]]
=> [[1,2,3,4,6,7,8,9,10,12],[5,11]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,2,3,4,5,6,8,10],[7,9]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,2,3,5,7,8,10],[4,6,9]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2,4,5,6,7,8,10],[3,9]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,2,3,5,6,9,10],[4,7,8]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,2,3,4,7,9,10],[5,6,8]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2,4,5,7,9,10],[3,6,8]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> [[1,2,4,5,6,7,8,10,11,12],[3,9]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2,4,6,7,8,9,10],[3,5]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> [[1,2,4,5,6,8,9,10,11,12],[3,7]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> [[1,2,4,5,6,7,8,10,11,12,13,14],[3,9]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> [[1,2,4,5,6,7,8,9,10,11,12,13,14,15,16],[3]]
=> ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772}
Description
Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition λ has dimension equal to the number of standard tableaux of shape λ. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape λ; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St000338
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000338: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2 = 3 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {6,10} - 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {6,10} - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,3,6,30,35} - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {1,3,6,30,35} - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? ∊ {1,3,6,30,35} - 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] => ? ∊ {1,3,6,30,35} - 1
[1,1,1,1]
=> [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] => ? ∊ {1,3,6,30,35} - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,6,6,30,30,126,140} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,6,6,30,30,126,140} - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {1,6,6,30,30,126,140} - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {1,6,6,30,30,126,140} - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {1,6,6,30,30,126,140} - 1
[2,1,1,1]
=> [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] => ? ∊ {1,6,6,30,30,126,140} - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,6,6,30,30,126,140} - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[3,1,1,1]
=> [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] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[3,2,1,1]
=> [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] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[2,2,2,1]
=> [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] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} - 1
Description
The number of pixed points of a permutation. For a permutation σ=pτ1τ2τk in its hook factorization, [1] defines pixσ=length(p).
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000501: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4 = 3 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ? ∊ {6,10} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? ∊ {6,10} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => ? ∊ {1,3,6,30,35} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ? ∊ {1,3,6,30,35} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => ? ∊ {1,3,6,30,35} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? ∊ {1,3,6,30,35} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,30,35} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {1,6,6,30,30,126,140} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => ? ∊ {1,6,6,30,30,126,140} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {1,6,6,30,30,126,140} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {1,6,6,30,30,126,140} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {1,6,6,30,30,126,140} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => ? ∊ {1,6,6,30,30,126,140} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,30,30,126,140} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [14,13,12,11,10,9,8,7,6,5,4,3,2,1,16,15] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [12,11,10,9,6,5,8,7,4,3,2,1,14,13] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [10,9,6,5,4,3,8,7,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [10,9,4,3,8,7,6,5,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,12,11,8,7,6,5,10,9,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,12,11,6,5,10,9,8,7,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,14,13,12,11,8,7,10,9,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,16,15,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
Description
The size of the first part in the decomposition of a permutation. For a permutation π of {1,,n}, this is defined to be the smallest k>0 such that {π(1),,π(k)}={1,,k}. This statistic is undefined for the empty permutation. For the number of parts in the decomposition see [[St000056]].
Matching statistic: St000542
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00058: Perfect matchings to permutationPermutations
St000542: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 12%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4 = 3 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ? ∊ {6,10} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? ∊ {6,10} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => ? ∊ {1,3,6,30,35} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ? ∊ {1,3,6,30,35} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => ? ∊ {1,3,6,30,35} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? ∊ {1,3,6,30,35} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,30,35} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => ? ∊ {1,6,6,30,30,126,140} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => ? ∊ {1,6,6,30,30,126,140} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {1,6,6,30,30,126,140} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {1,6,6,30,30,126,140} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {1,6,6,30,30,126,140} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => ? ∊ {1,6,6,30,30,126,140} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,30,30,126,140} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [10,9,8,5,4,7,6,3,2,1,12,11] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,12,11,10,7,6,9,8,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,3,6,6,10,30,60,140,210,462,630} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [14,13,12,11,10,9,8,7,6,5,4,3,2,1,16,15] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [12,11,10,9,6,5,8,7,4,3,2,1,14,13] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [10,9,6,5,4,3,8,7,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [10,9,4,3,8,7,6,5,2,1,12,11] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,12,11,8,7,6,5,10,9,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,12,11,6,5,10,9,8,7,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,14,13,12,11,8,7,10,9,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,16,15,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {1,6,6,6,30,30,30,60,140,140,420,630,1260,1716,2772} + 1
Description
The number of left-to-right-minima of a permutation. An integer σi in the one-line notation of a permutation σ is a left-to-right-minimum if there does not exist a j < i such that σj<σi.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000664The number of right ropes of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000990The first ascent of a permutation. St001058The breadth of the ordered tree. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001556The number of inversions of the third entry of a permutation. St001893The flag descent of a signed permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St000756The sum of the positions of the left to right maxima of a permutation.