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Your data matches 53 different statistics following compositions of up to 3 maps.
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Matching statistic: St001763
St001763: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 8
[1,1,1]
=> 24
[4]
=> 16
[3,1]
=> 81
[2,2]
=> 96
[2,1,1]
=> 480
[1,1,1,1]
=> 2880
[5]
=> 125
[4,1]
=> 1024
[3,2]
=> 1296
[3,1,1]
=> 9720
[2,2,1]
=> 11520
[2,1,1,1]
=> 100800
[1,1,1,1,1]
=> 1008000
[6]
=> 1296
[5,1]
=> 15625
[4,2]
=> 20480
[4,1,1]
=> 215040
[3,3]
=> 21870
[3,2,1]
=> 272160
[3,1,1,1]
=> 3265920
[2,2,2]
=> 322560
[2,2,1,1]
=> 3870720
[2,1,1,1,1]
=> 52254720
[1,1,1,1,1,1]
=> 783820800
Description
The Hurwitz number of an integer partition.
See [eq.(9),pg.21, 1].
Matching statistic: St000232
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> ? ∊ {8,24}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> ? ∊ {8,24}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> {{1,2,3,4,9},{5,6,7,8,10}}
=> ? ∊ {16,81,96,480,2880}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> ? ∊ {16,81,96,480,2880}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> ? ∊ {16,81,96,480,2880}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> ? ∊ {16,81,96,480,2880}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> {{1,3,4,5,6},{2,7,8,9,10}}
=> ? ∊ {16,81,96,480,2880}
[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]]
=> {{1,2,3,4,5,11},{6,7,8,9,10,12}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> {{1,2,3,5,9},{4,6,7,8,10}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> {{1,3,4,5,7},{2,6,8,9,10}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[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]]
=> {{1,3,4,5,6,7},{2,8,9,10,11,12}}
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[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]]
=> {{1,2,3,4,5,6,13},{7,8,9,10,11,12,14}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[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]]
=> {{1,2,3,4,6,11},{5,7,8,9,10,12}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> {{1,2,3,6,9},{4,5,7,8,10}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> {{1,2,4,5,9},{3,6,7,8,10}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> {{1,2,3,7,8},{4,5,6,9,10}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> {{1,3,4,5,8},{2,6,7,9,10}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> {{1,2,5,6,7},{3,4,8,9,10}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> {{1,3,4,6,7},{2,5,8,9,10}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[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]]
=> {{1,3,4,5,6,8},{2,7,9,10,11,12}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[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]]
=> {{1,3,4,5,6,7,8},{2,9,10,11,12,13,14}}
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St001582
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ? ∊ {8,24}
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ? ∊ {8,24}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {16,81,96,480,2880}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => ? ∊ {16,81,96,480,2880}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ? ∊ {16,81,96,480,2880}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => ? ∊ {16,81,96,480,2880}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ? ∊ {16,81,96,480,2880}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,3,2,6,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,3,5,2,6,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,5,4,3,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,8,7,6,5,4,3,1] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001625
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2]
=> [1,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {8,24}
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? ∊ {8,24}
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {16,81,96,480,2880}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {16,81,96,480,2880}
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? ∊ {16,81,96,480,2880}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {16,81,96,480,2880}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? ∊ {16,81,96,480,2880}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? ∊ {125,1024,1296,9720,11520,100800,1008000}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ?
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,14),(3,14),(4,9),(5,8),(6,8),(6,10),(7,9),(7,10),(8,11),(9,12),(10,11),(10,12),(11,13),(12,13),(13,14)],15)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,14),(3,14),(4,9),(5,8),(6,8),(6,10),(7,9),(7,10),(8,11),(9,12),(10,11),(10,12),(11,13),(12,13),(13,14)],15)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,17),(2,16),(3,15),(4,14),(4,17),(5,15),(5,19),(6,16),(6,20),(7,19),(7,20),(9,14),(10,9),(11,9),(12,10),(13,11),(14,8),(15,12),(16,13),(17,8),(18,10),(18,11),(19,12),(19,18),(20,13),(20,18)],21)
=> ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800}
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St000357
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000357: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000357: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
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)]
=> [4,3,2,1,6,5] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 0 = 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] => ? ∊ {8,24} - 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 3 - 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] => ? ∊ {8,24} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
Description
The number of occurrences of the pattern 12-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $12\!\!-\!\!3$.
Matching statistic: St000372
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
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)]
=> [4,3,2,1,6,5] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 0 = 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] => ? ∊ {8,24} - 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 3 - 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] => ? ∊ {8,24} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {16,81,96,480,2880} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Matching statistic: St000489
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000489: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000489: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => ? ∊ {8,24} - 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ? ∊ {8,24} - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [5,6,7,8,10,1,2,3,4,9] => ? ∊ {16,81,96,480,2880} - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => ? ∊ {16,81,96,480,2880} - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => ? ∊ {16,81,96,480,2880} - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => ? ∊ {16,81,96,480,2880} - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,7,8,9,10,1,3,4,5,6] => ? ∊ {16,81,96,480,2880} - 1
[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]]
=> [6,7,8,9,10,12,1,2,3,4,5,11] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [4,6,7,8,10,1,2,3,5,9] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,6,8,9,10,1,3,4,5,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[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]]
=> [2,8,9,10,11,12,1,3,4,5,6,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} - 1
[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]]
=> [7,8,9,10,11,12,14,1,2,3,4,5,6,13] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[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]]
=> [5,7,8,9,10,12,1,2,3,4,6,11] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [4,5,7,8,10,1,2,3,6,9] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [3,6,7,8,10,1,2,4,5,9] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [4,5,6,9,10,1,2,3,7,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [3,4,8,9,10,1,2,5,6,7] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [2,5,8,9,10,1,3,4,6,7] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[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]]
=> [2,7,9,10,11,12,1,3,4,5,6,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
[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]]
=> [2,9,10,11,12,13,14,1,3,4,5,6,7,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} - 1
Description
The number of cycles of a permutation of length at most 3.
Matching statistic: St000646
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => ? ∊ {8,24} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 4 = 3 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ? ∊ {8,24} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [5,6,7,8,10,1,2,3,4,9] => ? ∊ {16,81,96,480,2880} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => ? ∊ {16,81,96,480,2880} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => ? ∊ {16,81,96,480,2880} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => ? ∊ {16,81,96,480,2880} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,7,8,9,10,1,3,4,5,6] => ? ∊ {16,81,96,480,2880} + 1
[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]]
=> [6,7,8,9,10,12,1,2,3,4,5,11] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [4,6,7,8,10,1,2,3,5,9] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,6,8,9,10,1,3,4,5,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[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]]
=> [2,8,9,10,11,12,1,3,4,5,6,7] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 1
[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]]
=> [7,8,9,10,11,12,14,1,2,3,4,5,6,13] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[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]]
=> [5,7,8,9,10,12,1,2,3,4,6,11] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [4,5,7,8,10,1,2,3,6,9] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [3,6,7,8,10,1,2,4,5,9] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [4,5,6,9,10,1,2,3,7,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [3,4,8,9,10,1,2,5,6,7] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [2,5,8,9,10,1,3,4,6,7] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[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]]
=> [2,7,9,10,11,12,1,3,4,5,6,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
[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]]
=> [2,9,10,11,12,13,14,1,3,4,5,6,7,8] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see [[St000441]].
Matching statistic: St000837
(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
St000837: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
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] => 2 = 1 + 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] => ? ∊ {8,24} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4 = 3 + 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] => ? ∊ {8,24} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St001298
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
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] => 2 = 1 + 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] => ? ∊ {8,24} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4 = 3 + 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] => ? ∊ {8,24} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {16,81,96,480,2880} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {125,1024,1296,9720,11520,100800,1008000} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 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] => ? ∊ {1296,15625,20480,21870,215040,272160,322560,3265920,3870720,52254720,783820800} + 1
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St000037The sign of a permutation. St000490The intertwining number of a set partition. St000691The number of changes of a binary word. St000717The number of ordinal summands of a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001388The number of non-attacking neighbors of a permutation. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000638The number of up-down runs of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001439The number of even weak deficiencies and of odd weak exceedences. St000632The jump number of the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.
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