Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000863
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1] => [1] => 1
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 2
[1,1]
 => [1,1,0,0]
 => [2,1] => [2,1] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => 3
[2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => 3
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => 3
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => 4
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 5
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,5,2,3,4] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [5,1,2,3,6,4] => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [1,3,4,2] => 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [4,5,1,2,3] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [4,1,2,5,6,3] => 4
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [3,4,1,5,2] => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [3,1,4,5,6,2] => 5
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 5
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,5,3] => 3
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [5,6,1,2,3,4] => 4
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,4,5,1,2] => 4
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,5,3,4] => 4
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [4,5,1,2,6,3] => 4
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [4,1,5,2,3] => 3
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [3,4,1,5,6,2] => 5
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [1,5,2,3,6,4] => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [1,3,4,5,2] => 4
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => [4,5,1,6,2,3] => 4
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 5
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [1,3,2,5,4] => 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => [3,4,5,1,6,2] => 5
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => [5,1,6,2,3,4] => 4
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [1,2,4,5,3] => 4
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => [4,1,5,2,6,3] => 4
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,4,5,6,2,3] => [1,4,2,5,6,3] => 4
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => 6
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [3,4,5,6,1,2] => 5
Description
The length of the first row of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the length of the first row of $P$ and $Q$.
Matching statistic: St000718
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 71%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 3 = 1 + 2
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 2
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 2
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 5 + 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 5 + 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 6 + 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 5 + 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 6 + 2
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? = 7 + 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3 + 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 + 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 + 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3 + 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 4 + 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 5 + 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 5 + 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 4 + 2
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 2
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 4 + 2
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [2,5,4,1,7,3,6] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 6 + 2
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [6,3,4,1,2,7,5] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 5 + 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? = 4 + 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 4 + 2
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 4 + 2
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 5 + 2
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 2
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 5 + 2
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 4 + 2
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 2
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 5 + 2
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,8,1,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 5 + 2
Description
The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]
Matching statistic: St000834
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 86%
Values
[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1 = 2 - 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2 = 3 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? = 3 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? = 3 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 4 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 4 = 5 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? = 4 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? = 3 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? = 3 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 3 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? = 4 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? = 5 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 5 = 6 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 5 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? = 4 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? = 4 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? = 3 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? = 4 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? = 3 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? = 4 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? = 5 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 6 - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 7 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? = 5 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? = 3 - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? = 4 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? = 4 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? = 4 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 4 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? = 3 - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 5 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => ? = 4 - 1
[4,4]
 => [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] => ? = 4 - 1
[4,3,1]
 => [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] => ? = 4 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 5 - 1
[3,3,2]
 => [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] => ? = 3 - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => [4,6,7,3,9,5,2,11,8,12,10,1] => ? = 5 - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,7,8,9,11,6,5,4,12,10,3] => ? = 4 - 1
[2,2,2,2]
 => [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] => ? = 4 - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => [5,6,7,9,4,3,2,11,8,12,10,1] => ? = 4 - 1
[5,4]
 => [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] => ? = 4 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
 => [2,1,4,3,6,5,11,12,13,14,10,9,8,7] => ? = 6 - 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
 => [4,6,8,3,9,5,11,7,2,12,10,1] => ? = 5 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
 => [2,1,6,9,10,5,11,12,8,7,4,3] => ? = 4 - 1
[3,3,3]
 => [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] => ? = 4 - 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,4),(5,8),(6,7),(10,11)]
 => [4,7,8,3,9,11,6,5,2,12,10,1] => ? = 4 - 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,7,9,10,11,6,12,8,5,4,3] => ? = 4 - 1
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
 => [5,7,8,9,4,11,6,3,2,12,10,1] => ? = 4 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
 => [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 5 - 1
[4,4,2]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [(1,12),(2,11),(3,4),(5,6),(7,10),(8,9)]
 => [4,6,9,3,10,5,11,12,8,7,2,1] => ? = 4 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,8,9,10,11,12,7,6,5,4,3] => ? = 5 - 1
[4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
 => [2,1,4,3,9,11,12,13,8,14,10,7,6,5] => ? = 5 - 1
[3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
 => [6,7,8,9,11,5,4,3,2,12,10,1] => ? = 4 - 1
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 $[w_1,..., w_n,0]$.