Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000354
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,5,3,2,4] => 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,2,3] => 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,4,2,5,3] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,4,5,3,2] => 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,2,4] => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,4,3,2,5] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,3,2,4] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,5,3,2,6,4] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,3,2,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,5,4,2,3] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,4,2,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,2,6,5,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,6,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,2,6,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,5,2,3,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,2,3,6,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,2,4,6,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,4,3,5] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,3,4] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,3,6,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,5,4,2] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,4,2,5] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,6,5,3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,6,4,5,3,2] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,6,3,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,5,2,6,4] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,5,6,3,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => 2
Description
The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000245
(load all 3 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? = 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] => [2,1,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 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] => [4,3,2,1,8,7,6,5] => 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] => [6,5,3,4,2,1,8,7] => ? = 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 2
[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] => [2,1,10,7,5,9,4,8,6,3] => ? = 2
[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] => [2,1,10,9,5,8,7,6,4,3] => ? = 2
[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] => [2,1,8,7,6,5,4,3,10,9] => 2
[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] => [2,1,10,7,9,6,4,8,5,3] => ? = 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] => [2,1,10,9,8,6,7,5,4,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 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] => [4,3,2,1,10,9,7,8,6,5] => ? = 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 3
[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] => [6,5,3,4,2,1,10,9,8,7] => ? = 2
[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] => [8,5,3,7,2,6,4,1,10,9] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 3
[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] => [8,7,3,6,5,4,2,1,10,9] => ? = 3
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 2
[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] => [6,5,4,3,2,1,8,7,10,9] => 2
[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] => [6,5,4,3,2,1,10,9,8,7] => 1
[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] => [8,5,7,4,2,6,3,1,10,9] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 2
[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] => [8,7,6,4,5,3,2,1,10,9] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 2
[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] => [8,7,6,5,4,3,2,1,10,9] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => [2,1,12,9,8,6,11,5,4,10,7,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,6,8,10,5,11,7,12,9,4,3] => [2,1,12,11,10,8,7,6,9,5,4,3] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [3,4,2,1,6,5,9,11,8,12,10,7] => [4,3,2,1,6,5,12,11,9,10,8,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [3,4,2,1,6,5,10,11,12,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [3,4,2,1,7,8,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [3,4,2,1,7,8,6,5,11,12,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [3,4,2,1,7,9,6,10,8,5,12,11] => [4,3,2,1,10,9,7,8,6,5,12,11] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [3,4,2,1,7,9,6,11,8,12,10,5] => [4,3,2,1,12,9,7,11,6,10,8,5] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [3,4,2,1,7,10,6,11,12,9,8,5] => [4,3,2,1,12,11,7,10,9,8,6,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [3,4,2,1,8,9,10,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [3,4,2,1,8,9,11,7,6,12,10,5] => [4,3,2,1,12,9,11,8,6,10,7,5] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [3,4,2,1,8,10,11,7,12,9,6,5] => [4,3,2,1,12,11,10,8,9,7,6,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [3,4,2,1,9,10,11,12,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [3,5,2,6,4,1,8,7,10,9,12,11] => [6,5,3,4,2,1,8,7,10,9,12,11] => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [4,5,6,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 3
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [4,5,6,3,2,1,8,7,11,12,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [4,5,6,3,2,1,9,10,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [4,5,6,3,2,1,10,11,12,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [5,6,7,8,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [5,6,7,8,4,3,2,1,11,12,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? = 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] => [2,1,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 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] => [4,3,2,1,8,7,6,5] => 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] => [6,5,3,4,2,1,8,7] => ? = 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 2
[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] => [2,1,10,7,5,9,4,8,6,3] => ? = 2
[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] => [2,1,10,9,5,8,7,6,4,3] => ? = 2
[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] => [2,1,8,7,6,5,4,3,10,9] => 2
[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] => [2,1,10,7,9,6,4,8,5,3] => ? = 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] => [2,1,10,9,8,6,7,5,4,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 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] => [4,3,2,1,10,9,7,8,6,5] => ? = 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 3
[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] => [6,5,3,4,2,1,10,9,8,7] => ? = 2
[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] => [8,5,3,7,2,6,4,1,10,9] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 3
[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] => [8,7,3,6,5,4,2,1,10,9] => ? = 3
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 2
[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] => [6,5,4,3,2,1,8,7,10,9] => 2
[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] => [6,5,4,3,2,1,10,9,8,7] => 1
[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] => [8,5,7,4,2,6,3,1,10,9] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 2
[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] => [8,7,6,4,5,3,2,1,10,9] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 2
[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] => [8,7,6,5,4,3,2,1,10,9] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => [2,1,12,9,8,6,11,5,4,10,7,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,6,8,10,5,11,7,12,9,4,3] => [2,1,12,11,10,8,7,6,9,5,4,3] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [3,4,2,1,6,5,9,11,8,12,10,7] => [4,3,2,1,6,5,12,11,9,10,8,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [3,4,2,1,6,5,10,11,12,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [3,4,2,1,7,8,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [3,4,2,1,7,8,6,5,11,12,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [3,4,2,1,7,9,6,10,8,5,12,11] => [4,3,2,1,10,9,7,8,6,5,12,11] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [3,4,2,1,7,9,6,11,8,12,10,5] => [4,3,2,1,12,9,7,11,6,10,8,5] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [3,4,2,1,7,10,6,11,12,9,8,5] => [4,3,2,1,12,11,7,10,9,8,6,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [3,4,2,1,8,9,10,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [3,4,2,1,8,9,11,7,6,12,10,5] => [4,3,2,1,12,9,11,8,6,10,7,5] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [3,4,2,1,8,10,11,7,12,9,6,5] => [4,3,2,1,12,11,10,8,9,7,6,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [3,4,2,1,9,10,11,12,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [3,5,2,6,4,1,8,7,10,9,12,11] => [6,5,3,4,2,1,8,7,10,9,12,11] => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [4,5,6,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 3
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [4,5,6,3,2,1,8,7,11,12,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [4,5,6,3,2,1,9,10,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [4,5,6,3,2,1,10,11,12,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [5,6,7,8,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [5,6,7,8,4,3,2,1,11,12,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is $$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$ for some $(r,a,b)$. This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000834
(load all 3 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? = 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] => [2,1,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 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] => [4,3,2,1,8,7,6,5] => 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] => [6,5,3,4,2,1,8,7] => ? = 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 2
[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] => [2,1,10,7,5,9,4,8,6,3] => ? = 2
[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] => [2,1,10,9,5,8,7,6,4,3] => ? = 2
[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] => [2,1,8,7,6,5,4,3,10,9] => 2
[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] => [2,1,10,7,9,6,4,8,5,3] => ? = 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] => [2,1,10,9,8,6,7,5,4,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 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] => [4,3,2,1,10,9,7,8,6,5] => ? = 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 3
[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] => [6,5,3,4,2,1,10,9,8,7] => ? = 2
[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] => [8,5,3,7,2,6,4,1,10,9] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 3
[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] => [8,7,3,6,5,4,2,1,10,9] => ? = 3
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 2
[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] => [6,5,4,3,2,1,8,7,10,9] => 2
[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] => [6,5,4,3,2,1,10,9,8,7] => 1
[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] => [8,5,7,4,2,6,3,1,10,9] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 2
[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] => [8,7,6,4,5,3,2,1,10,9] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 2
[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] => [8,7,6,5,4,3,2,1,10,9] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => [2,1,12,9,8,6,11,5,4,10,7,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,6,8,10,5,11,7,12,9,4,3] => [2,1,12,11,10,8,7,6,9,5,4,3] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [3,4,2,1,6,5,9,11,8,12,10,7] => [4,3,2,1,6,5,12,11,9,10,8,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [3,4,2,1,6,5,10,11,12,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [3,4,2,1,7,8,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [3,4,2,1,7,8,6,5,11,12,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [3,4,2,1,7,9,6,10,8,5,12,11] => [4,3,2,1,10,9,7,8,6,5,12,11] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [3,4,2,1,7,9,6,11,8,12,10,5] => [4,3,2,1,12,9,7,11,6,10,8,5] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [3,4,2,1,7,10,6,11,12,9,8,5] => [4,3,2,1,12,11,7,10,9,8,6,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [3,4,2,1,8,9,10,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [3,4,2,1,8,9,11,7,6,12,10,5] => [4,3,2,1,12,9,11,8,6,10,7,5] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [3,4,2,1,8,10,11,7,12,9,6,5] => [4,3,2,1,12,11,10,8,9,7,6,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [3,4,2,1,9,10,11,12,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [3,5,2,6,4,1,8,7,10,9,12,11] => [6,5,3,4,2,1,8,7,10,9,12,11] => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [4,5,6,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 3
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [4,5,6,3,2,1,8,7,11,12,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [4,5,6,3,2,1,9,10,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [4,5,6,3,2,1,10,11,12,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [5,6,7,8,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [5,6,7,8,4,3,2,1,11,12,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of right outer peaks of a permutation. A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000871
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [5,4,1,6,3,2] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => ? = 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] => [2,1,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 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] => [4,3,2,1,8,7,6,5] => 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] => [5,4,1,6,3,2,8,7] => ? = 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? = 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? = 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? = 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? = 2
[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] => [2,1,9,8,3,7,6,10,5,4] => ? = 2
[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] => [2,1,9,8,7,6,3,10,5,4] => ? = 2
[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] => [2,1,8,7,6,5,4,3,10,9] => 2
[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] => [2,1,9,8,3,10,7,6,5,4] => ? = 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] => [2,1,8,7,4,3,10,9,6,5] => ? = 2
[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] => [2,1,10,9,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 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] => [4,3,2,1,9,8,5,10,7,6] => ? = 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? = 3
[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] => [5,4,1,6,3,2,10,9,8,7] => ? = 2
[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] => [7,6,1,5,4,8,3,2,10,9] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? = 3
[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] => [7,6,5,4,1,8,3,2,10,9] => ? = 3
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? = 2
[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] => [6,5,4,3,2,1,8,7,10,9] => 2
[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] => [6,5,4,3,2,1,10,9,8,7] => 1
[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] => [7,6,1,8,5,4,3,2,10,9] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? = 2
[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] => [6,5,2,1,8,7,4,3,10,9] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? = 2
[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] => [8,7,6,5,4,3,2,1,10,9] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,12,9,6,5,8,7,4,11,10,3] => [2,1,11,10,3,8,7,4,12,9,6,5] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,12,11,6,5,8,7,10,9,4,3] => [2,1,10,9,4,3,8,7,12,11,6,5] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [4,3,2,1,6,5,10,9,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [4,3,2,1,6,5,12,9,8,11,10,7] => [4,3,2,1,6,5,11,10,7,12,9,8] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [4,3,2,1,6,5,12,11,10,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [4,3,2,1,8,7,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [4,3,2,1,8,7,6,5,12,11,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [4,3,2,1,10,7,6,9,8,5,12,11] => [4,3,2,1,9,8,5,10,7,6,12,11] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [4,3,2,1,12,7,6,9,8,11,10,5] => [4,3,2,1,11,10,5,9,8,12,7,6] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [4,3,2,1,12,7,6,11,10,9,8,5] => [4,3,2,1,11,10,9,8,5,12,7,6] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [4,3,2,1,10,9,8,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [4,3,2,1,12,9,8,7,6,11,10,5] => [4,3,2,1,11,10,5,12,9,8,7,6] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [4,3,2,1,12,11,8,7,10,9,6,5] => [4,3,2,1,10,9,6,5,12,11,8,7] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [4,3,2,1,12,11,10,9,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [6,3,2,5,4,1,8,7,10,9,12,11] => [5,4,1,6,3,2,8,7,10,9,12,11] => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [6,5,4,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 3
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [6,5,4,3,2,1,8,7,12,11,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [6,5,4,3,2,1,10,9,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [6,5,4,3,2,1,12,11,10,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [8,7,6,5,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [8,7,6,5,4,3,2,1,12,11,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of very big ascents of a permutation. A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$. For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St001086
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [5,4,1,6,3,2] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => ? = 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] => [2,1,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 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] => [4,3,2,1,8,7,6,5] => 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] => [5,4,1,6,3,2,8,7] => ? = 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? = 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? = 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? = 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? = 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? = 2
[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] => [2,1,9,8,3,7,6,10,5,4] => ? = 2
[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] => [2,1,9,8,7,6,3,10,5,4] => ? = 2
[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] => [2,1,8,7,6,5,4,3,10,9] => 2
[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] => [2,1,9,8,3,10,7,6,5,4] => ? = 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] => [2,1,8,7,4,3,10,9,6,5] => ? = 2
[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] => [2,1,10,9,8,7,6,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 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] => [4,3,2,1,9,8,5,10,7,6] => ? = 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? = 3
[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] => [5,4,1,6,3,2,10,9,8,7] => ? = 2
[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] => [7,6,1,5,4,8,3,2,10,9] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? = 3
[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] => [7,6,5,4,1,8,3,2,10,9] => ? = 3
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? = 2
[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] => [6,5,4,3,2,1,8,7,10,9] => 2
[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] => [6,5,4,3,2,1,10,9,8,7] => 1
[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] => [7,6,1,8,5,4,3,2,10,9] => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? = 2
[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] => [6,5,2,1,8,7,4,3,10,9] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? = 2
[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] => [8,7,6,5,4,3,2,1,10,9] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,12,9,6,5,8,7,4,11,10,3] => [2,1,11,10,3,8,7,4,12,9,6,5] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,12,11,6,5,8,7,10,9,4,3] => [2,1,10,9,4,3,8,7,12,11,6,5] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [4,3,2,1,6,5,10,9,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [4,3,2,1,6,5,12,9,8,11,10,7] => [4,3,2,1,6,5,11,10,7,12,9,8] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [4,3,2,1,6,5,12,11,10,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [4,3,2,1,8,7,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [4,3,2,1,8,7,6,5,12,11,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [4,3,2,1,10,7,6,9,8,5,12,11] => [4,3,2,1,9,8,5,10,7,6,12,11] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [4,3,2,1,12,7,6,9,8,11,10,5] => [4,3,2,1,11,10,5,9,8,12,7,6] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [4,3,2,1,12,7,6,11,10,9,8,5] => [4,3,2,1,11,10,9,8,5,12,7,6] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [4,3,2,1,10,9,8,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [4,3,2,1,12,9,8,7,6,11,10,5] => [4,3,2,1,11,10,5,12,9,8,7,6] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [4,3,2,1,12,11,8,7,10,9,6,5] => [4,3,2,1,10,9,6,5,12,11,8,7] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [4,3,2,1,12,11,10,9,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [6,3,2,5,4,1,8,7,10,9,12,11] => [5,4,1,6,3,2,8,7,10,9,12,11] => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [6,5,4,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 3
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [6,5,4,3,2,1,8,7,12,11,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 2
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [6,5,4,3,2,1,10,9,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 2
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [6,5,4,3,2,1,12,11,10,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [8,7,6,5,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [8,7,6,5,4,3,2,1,12,11,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000035
(load all 3 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1 = 0 + 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] => [2,1,4,3,6,5,8,7] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 3 = 2 + 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] => [2,1,6,5,4,3,8,7] => 3 = 2 + 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] => [2,1,8,7,5,6,4,3] => ? = 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] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 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] => [6,5,3,4,2,1,8,7] => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 1 + 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 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] => [2,1,4,3,10,9,8,7,6,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 2 + 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] => [2,1,10,7,5,9,4,8,6,3] => ? = 2 + 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] => [2,1,10,9,5,8,7,6,4,3] => ? = 2 + 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] => [2,1,8,7,6,5,4,3,10,9] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => [2,1,10,7,9,6,4,8,5,3] => ? = 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] => [2,1,10,9,8,6,7,5,4,3] => ? = 2 + 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] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3 = 2 + 1
[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] => [4,3,2,1,10,9,7,8,6,5] => ? = 1 + 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 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] => [6,5,3,4,2,1,10,9,8,7] => ? = 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] => [8,5,3,7,2,6,4,1,10,9] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 3 + 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] => [8,7,3,6,5,4,2,1,10,9] => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 2 + 1
[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] => [6,5,4,3,2,1,8,7,10,9] => 3 = 2 + 1
[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] => [6,5,4,3,2,1,10,9,8,7] => 2 = 1 + 1
[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] => [8,5,7,4,2,6,3,1,10,9] => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 2 + 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] => [8,7,6,4,5,3,2,1,10,9] => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 2 + 1
[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] => [8,7,6,5,4,3,2,1,10,9] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => [2,1,12,9,8,6,11,5,4,10,7,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,6,8,10,5,11,7,12,9,4,3] => [2,1,12,11,10,8,7,6,9,5,4,3] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [3,4,2,1,6,5,9,11,8,12,10,7] => [4,3,2,1,6,5,12,11,9,10,8,7] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [3,4,2,1,6,5,10,11,12,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [3,4,2,1,7,8,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [3,4,2,1,7,8,6,5,11,12,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [3,4,2,1,7,9,6,10,8,5,12,11] => [4,3,2,1,10,9,7,8,6,5,12,11] => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [3,4,2,1,7,9,6,11,8,12,10,5] => [4,3,2,1,12,9,7,11,6,10,8,5] => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [3,4,2,1,7,10,6,11,12,9,8,5] => [4,3,2,1,12,11,7,10,9,8,6,5] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [3,4,2,1,8,9,10,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [3,4,2,1,8,9,11,7,6,12,10,5] => [4,3,2,1,12,9,11,8,6,10,7,5] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [3,4,2,1,8,10,11,7,12,9,6,5] => [4,3,2,1,12,11,10,8,9,7,6,5] => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [3,4,2,1,9,10,11,12,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [3,5,2,6,4,1,8,7,10,9,12,11] => [6,5,3,4,2,1,8,7,10,9,12,11] => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [4,5,6,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [4,5,6,3,2,1,8,7,11,12,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [4,5,6,3,2,1,9,10,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [4,5,6,3,2,1,10,11,12,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [5,6,7,8,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [5,6,7,8,4,3,2,1,11,12,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 2 = 1 + 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
Description
The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000153
(load all 2 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1 = 0 + 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] => [2,1,4,3,6,5,8,7] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 3 = 2 + 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] => [2,1,6,5,4,3,8,7] => 3 = 2 + 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] => [2,1,8,7,5,6,4,3] => ? = 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] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 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] => [6,5,3,4,2,1,8,7] => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 1 + 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 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] => [2,1,4,3,10,9,8,7,6,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 2 + 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] => [2,1,10,7,5,9,4,8,6,3] => ? = 2 + 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] => [2,1,10,9,5,8,7,6,4,3] => ? = 2 + 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] => [2,1,8,7,6,5,4,3,10,9] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => [2,1,10,7,9,6,4,8,5,3] => ? = 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] => [2,1,10,9,8,6,7,5,4,3] => ? = 2 + 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] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3 = 2 + 1
[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] => [4,3,2,1,10,9,7,8,6,5] => ? = 1 + 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 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] => [6,5,3,4,2,1,10,9,8,7] => ? = 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] => [8,5,3,7,2,6,4,1,10,9] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 3 + 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] => [8,7,3,6,5,4,2,1,10,9] => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 2 + 1
[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] => [6,5,4,3,2,1,8,7,10,9] => 3 = 2 + 1
[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] => [6,5,4,3,2,1,10,9,8,7] => 2 = 1 + 1
[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] => [8,5,7,4,2,6,3,1,10,9] => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 2 + 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] => [8,7,6,4,5,3,2,1,10,9] => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 2 + 1
[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] => [8,7,6,5,4,3,2,1,10,9] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => [2,1,12,9,8,6,11,5,4,10,7,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,6,8,10,5,11,7,12,9,4,3] => [2,1,12,11,10,8,7,6,9,5,4,3] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [3,4,2,1,6,5,9,11,8,12,10,7] => [4,3,2,1,6,5,12,11,9,10,8,7] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [3,4,2,1,6,5,10,11,12,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [3,4,2,1,7,8,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [3,4,2,1,7,8,6,5,11,12,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [3,4,2,1,7,9,6,10,8,5,12,11] => [4,3,2,1,10,9,7,8,6,5,12,11] => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [3,4,2,1,7,9,6,11,8,12,10,5] => [4,3,2,1,12,9,7,11,6,10,8,5] => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [3,4,2,1,7,10,6,11,12,9,8,5] => [4,3,2,1,12,11,7,10,9,8,6,5] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [3,4,2,1,8,9,10,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [3,4,2,1,8,9,11,7,6,12,10,5] => [4,3,2,1,12,9,11,8,6,10,7,5] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [3,4,2,1,8,10,11,7,12,9,6,5] => [4,3,2,1,12,11,10,8,9,7,6,5] => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [3,4,2,1,9,10,11,12,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [3,5,2,6,4,1,8,7,10,9,12,11] => [6,5,3,4,2,1,8,7,10,9,12,11] => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [4,5,6,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [4,5,6,3,2,1,8,7,11,12,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [4,5,6,3,2,1,9,10,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [4,5,6,3,2,1,10,11,12,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [5,6,7,8,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [5,6,7,8,4,3,2,1,11,12,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 2 = 1 + 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
Description
The number of adjacent cycles of a permutation. This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000374
(load all 2 compositions to match this statistic)
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [5,4,1,6,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1 = 0 + 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] => [2,1,4,3,6,5,8,7] => 4 = 3 + 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] => [2,1,4,3,8,7,6,5] => 3 = 2 + 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] => [2,1,6,5,4,3,8,7] => 3 = 2 + 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] => [2,1,7,6,3,8,5,4] => ? = 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] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 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] => [5,4,1,6,3,2,8,7] => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? = 1 + 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? = 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] => [2,1,4,3,10,9,8,7,6,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? = 2 + 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] => [2,1,9,8,3,7,6,10,5,4] => ? = 2 + 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] => [2,1,9,8,7,6,3,10,5,4] => ? = 2 + 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] => [2,1,8,7,6,5,4,3,10,9] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => [2,1,9,8,3,10,7,6,5,4] => ? = 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] => [2,1,8,7,4,3,10,9,6,5] => ? = 2 + 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] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3 = 2 + 1
[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] => [4,3,2,1,9,8,5,10,7,6] => ? = 1 + 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? = 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] => [5,4,1,6,3,2,10,9,8,7] => ? = 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] => [7,6,1,5,4,8,3,2,10,9] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? = 3 + 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] => [7,6,5,4,1,8,3,2,10,9] => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? = 2 + 1
[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] => [6,5,4,3,2,1,8,7,10,9] => 3 = 2 + 1
[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] => [6,5,4,3,2,1,10,9,8,7] => 2 = 1 + 1
[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] => [7,6,1,8,5,4,3,2,10,9] => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? = 2 + 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] => [6,5,2,1,8,7,4,3,10,9] => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? = 2 + 1
[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] => [8,7,6,5,4,3,2,1,10,9] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,12,9,6,5,8,7,4,11,10,3] => [2,1,11,10,3,8,7,4,12,9,6,5] => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,12,11,6,5,8,7,10,9,4,3] => [2,1,10,9,4,3,8,7,12,11,6,5] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [4,3,2,1,6,5,10,9,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [4,3,2,1,6,5,12,9,8,11,10,7] => [4,3,2,1,6,5,11,10,7,12,9,8] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [4,3,2,1,6,5,12,11,10,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [4,3,2,1,8,7,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [4,3,2,1,8,7,6,5,12,11,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [4,3,2,1,10,7,6,9,8,5,12,11] => [4,3,2,1,9,8,5,10,7,6,12,11] => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [4,3,2,1,12,7,6,9,8,11,10,5] => [4,3,2,1,11,10,5,9,8,12,7,6] => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [4,3,2,1,12,7,6,11,10,9,8,5] => [4,3,2,1,11,10,9,8,5,12,7,6] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [4,3,2,1,10,9,8,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [4,3,2,1,12,9,8,7,6,11,10,5] => [4,3,2,1,11,10,5,12,9,8,7,6] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [4,3,2,1,12,11,8,7,10,9,6,5] => [4,3,2,1,10,9,6,5,12,11,8,7] => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [4,3,2,1,12,11,10,9,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [6,3,2,5,4,1,8,7,10,9,12,11] => [5,4,1,6,3,2,8,7,10,9,12,11] => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [6,5,4,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [6,5,4,3,2,1,8,7,12,11,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [6,5,4,3,2,1,10,9,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [6,5,4,3,2,1,12,11,10,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [8,7,6,5,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [8,7,6,5,4,3,2,1,12,11,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 2 = 1 + 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
Description
The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
Matching statistic: St000742
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [5,4,1,6,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1 = 0 + 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] => [2,1,4,3,6,5,8,7] => 4 = 3 + 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] => [2,1,4,3,8,7,6,5] => 3 = 2 + 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] => [2,1,6,5,4,3,8,7] => 3 = 2 + 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] => [2,1,7,6,3,8,5,4] => ? = 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] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 2 = 1 + 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] => [5,4,1,6,3,2,8,7] => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? = 1 + 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? = 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] => [2,1,4,3,10,9,8,7,6,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? = 2 + 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] => [2,1,9,8,3,7,6,10,5,4] => ? = 2 + 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] => [2,1,9,8,7,6,3,10,5,4] => ? = 2 + 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] => [2,1,8,7,6,5,4,3,10,9] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => [2,1,9,8,3,10,7,6,5,4] => ? = 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] => [2,1,8,7,4,3,10,9,6,5] => ? = 2 + 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] => [2,1,10,9,8,7,6,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3 = 2 + 1
[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] => [4,3,2,1,9,8,5,10,7,6] => ? = 1 + 1
[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] => [4,3,2,1,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? = 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] => [5,4,1,6,3,2,10,9,8,7] => ? = 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] => [7,6,1,5,4,8,3,2,10,9] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? = 3 + 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] => [7,6,5,4,1,8,3,2,10,9] => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? = 2 + 1
[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] => [6,5,4,3,2,1,8,7,10,9] => 3 = 2 + 1
[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] => [6,5,4,3,2,1,10,9,8,7] => 2 = 1 + 1
[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] => [7,6,1,8,5,4,3,2,10,9] => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? = 2 + 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] => [6,5,2,1,8,7,4,3,10,9] => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? = 2 + 1
[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] => [8,7,6,5,4,3,2,1,10,9] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,12,9,6,5,8,7,4,11,10,3] => [2,1,11,10,3,8,7,4,12,9,6,5] => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,12,11,6,5,8,7,10,9,4,3] => [2,1,10,9,4,3,8,7,12,11,6,5] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [4,3,2,1,6,5,10,9,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
 => [4,3,2,1,6,5,12,9,8,11,10,7] => [4,3,2,1,6,5,11,10,7,12,9,8] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
 => [4,3,2,1,6,5,12,11,10,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
 => [4,3,2,1,8,7,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
 => [4,3,2,1,8,7,6,5,12,11,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
 => [4,3,2,1,10,7,6,9,8,5,12,11] => [4,3,2,1,9,8,5,10,7,6,12,11] => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
 => [4,3,2,1,12,7,6,9,8,11,10,5] => [4,3,2,1,11,10,5,9,8,12,7,6] => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [4,3,2,1,12,7,6,11,10,9,8,5] => [4,3,2,1,11,10,9,8,5,12,7,6] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
 => [4,3,2,1,10,9,8,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [4,3,2,1,12,9,8,7,6,11,10,5] => [4,3,2,1,11,10,5,12,9,8,7,6] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [4,3,2,1,12,11,8,7,10,9,6,5] => [4,3,2,1,10,9,6,5,12,11,8,7] => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [4,3,2,1,12,11,10,9,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
 => [6,3,2,5,4,1,8,7,10,9,12,11] => [5,4,1,6,3,2,8,7,10,9,12,11] => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
 => [6,5,4,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
 => [6,5,4,3,2,1,8,7,12,11,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [6,5,4,3,2,1,10,9,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [6,5,4,3,2,1,12,11,10,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [8,7,6,5,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [8,7,6,5,4,3,2,1,12,11,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 2 = 1 + 1
[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] => [10,9,8,7,6,5,4,3,2,1,12,11] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1 = 0 + 1
Description
The number of big ascents of a permutation after prepending zero. Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$. It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]). G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]]. [[St000646]] is the statistic without the border condition $\pi(0) = 0$.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000996The number of exclusive left-to-right maxima of a permutation. St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation.