searching the database
Your data matches 23 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
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 ℓ parts,
is fλ(1,1,…,1) where there are ℓ variables substituted by 1.
Matching statistic: St001632
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 14%●distinct values known / distinct values provided: 6%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
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 path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 14%●distinct values known / distinct values provided: 12%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
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 path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
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 2≤j≤n such that for all 1≤i<j, the pair (i,j) is an inversion of π.
Matching statistic: St001207
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
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 path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001778: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
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 path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard 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 path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000338: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
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=pτ1τ2⋯τk in its hook factorization, [1] defines
pixσ=length(p).
Matching statistic: St000501
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000501: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
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 path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!