- FindStat

Your data matches 112 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000260
(load all 4 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1] => ([],1)
 => 0
[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => 2
[[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 2
[[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[1,4],[2,5],[3],[6]]
 => [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 2
[[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[[1,3,5,6],[2],[4],[7]]
 => [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 3
[[1,3,6],[2,5],[4,7]]
 => [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 3
[[1,4,5],[2,7],[3],[6]]
 => [6,3,2,7,1,4,5] => [3,2,6,1,4,7,5] => ([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
 => 2
[[1,3,5],[2,7],[4],[6]]
 => [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => 2
[[1,3,6],[2,5],[4],[7]]
 => [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
 => 3
[[1,3,5],[2,6],[4],[7]]
 => [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
 => 3
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [4,3,7,2,1,5,6] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 2
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 2
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [3,2,7,6,1,4,5] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 2
[[1,4],[2,6],[3,7],[5]]
 => [5,3,7,2,6,1,4] => [3,5,2,1,7,6,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => 2
[[1,3],[2,6],[4,7],[5]]
 => [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 2
[[1,4],[2,5],[3,7],[6]]
 => [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => 2
[[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [2,4,6,1,7,5,3] => ([(0,5),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
 => 2
[[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [2,4,1,7,6,5,3] => ([(0,2),(1,2),(1,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [4,6,3,2,1,7,5] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [3,6,5,2,1,7,4] => ([(0,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [2,6,5,4,1,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 2
[[1,5],[2,6],[3],[4],[7]]
 => [7,4,3,2,6,1,5] => [4,3,7,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 2
[[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [3,5,7,2,1,6,4] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
 => 2
[[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [2,5,7,4,1,6,3] => ([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 2
[[1,4],[2,5],[3],[6],[7]]
 => [7,6,3,2,5,1,4] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 2
[[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [2,4,7,6,1,5,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 2
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [5,7,4,3,2,1,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [4,7,6,3,2,1,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [3,7,6,5,2,1,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [2,7,6,5,4,1,3] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St001928
(load all 2 compositions to match this statistic)

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [[1]]
 => [1] => [] => ? = 0 - 1
[[1],[2]]
 => [[1,2]]
 => [1,2] => [1] => 0 = 1 - 1
[[1],[2],[3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [4,1,2,3] => 1 = 2 - 1
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
 => [[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [5,4,1,2,3] => 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => [5,3,1,2,4] => 1 = 2 - 1
[[1,3],[2,5],[4,6]]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => [3,5,1,2,4] => 1 = 2 - 1
[[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[[1,4],[2,5],[3],[6]]
 => [[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [4,5,1,2,3] => 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
 => [[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [3,5,1,2,4] => 1 = 2 - 1
[[1,5],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4] => 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
 => [[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [4,1,2,3,5] => 1 = 2 - 1
[[1,3],[2],[4],[5],[6]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5] => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
 => [[1,2,4,7],[3],[5],[6]]
 => [6,5,3,1,2,4,7] => [6,5,3,1,2,4] => 2 = 3 - 1
[[1,3,6],[2,5],[4,7]]
 => [[1,2,4],[3,5,7],[6]]
 => [6,3,5,7,1,2,4] => [6,3,5,1,2,4] => 2 = 3 - 1
[[1,4,5],[2,7],[3],[6]]
 => [[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [5,4,1,2,3,6] => 1 = 2 - 1
[[1,3,5],[2,7],[4],[6]]
 => [[1,2,4,6],[3,7],[5]]
 => [5,3,7,1,2,4,6] => [5,3,1,2,4,6] => 1 = 2 - 1
[[1,3,6],[2,5],[4],[7]]
 => [[1,2,4,7],[3,5],[6]]
 => [6,3,5,1,2,4,7] => [6,3,5,1,2,4] => 2 = 3 - 1
[[1,3,5],[2,6],[4],[7]]
 => [[1,2,4,7],[3,6],[5]]
 => [5,3,6,1,2,4,7] => [5,3,6,1,2,4] => 2 = 3 - 1
[[1,5,6],[2],[3],[4],[7]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [6,5,1,2,3,4] => 1 = 2 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [6,4,1,2,3,5] => 1 = 2 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [6,3,1,2,4,5] => 1 = 2 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [5,4,1,2,3,6] => 1 = 2 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [5,3,1,2,4,6] => 1 = 2 - 1
[[1,4],[2,6],[3,7],[5]]
 => [[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [4,6,1,2,3,5] => 1 = 2 - 1
[[1,3],[2,6],[4,7],[5]]
 => [[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [3,6,1,2,4,5] => 1 = 2 - 1
[[1,4],[2,5],[3,7],[6]]
 => [[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => [4,5,1,2,3,6] => 1 = 2 - 1
[[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => [3,5,1,2,4,6] => 1 = 2 - 1
[[1,3],[2,5],[4,6],[7]]
 => [[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => [3,5,6,1,2,4] => 1 = 2 - 1
[[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [5,1,2,3,4,6] => 1 = 2 - 1
[[1,4],[2,7],[3],[5],[6]]
 => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [4,1,2,3,5,6] => 1 = 2 - 1
[[1,3],[2,7],[4],[5],[6]]
 => [[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [3,1,2,4,5,6] => 1 = 2 - 1
[[1,5],[2,6],[3],[4],[7]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4] => 1 = 2 - 1
[[1,4],[2,6],[3],[5],[7]]
 => [[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => [4,6,1,2,3,5] => 1 = 2 - 1
[[1,3],[2,6],[4],[5],[7]]
 => [[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => [3,6,1,2,4,5] => 1 = 2 - 1
[[1,4],[2,5],[3],[6],[7]]
 => [[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [4,5,1,2,3,6] => 1 = 2 - 1
[[1,3],[2,5],[4],[6],[7]]
 => [[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [3,5,1,2,4,6] => 1 = 2 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5] => 1 = 2 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [5,1,2,3,4,6] => 1 = 2 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => [4,1,2,3,5,6] => 1 = 2 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6] => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => 0 = 1 - 1

Description

The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence

$\pi_i,\pi_{i+1},\dots,\pi_k$ contributes

$\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.

Matching statistic: St000640
(load all 14 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000640: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 88%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1] => [1] => ([],1)
 => ? = 0 - 1
[[1],[2]]
 => [2,1] => [2,1] => ([],2)
 => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
 => 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [6,3,2,1,5,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [6,4,2,1,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => 1 = 2 - 1
[[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [5,3,2,6,1,4] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [5,4,2,6,1,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[1,4],[2,5],[3],[6]]
 => [6,3,2,5,1,4] => [6,3,2,5,1,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [6,4,2,5,1,3] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
 => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([],6)
 => 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
 => [7,4,2,1,3,5,6] => [7,4,2,1,6,5,3] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 3 - 1
[[1,3,6],[2,5],[4,7]]
 => [4,7,2,5,1,3,6] => [4,7,2,6,1,5,3] => ([(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6)],7)
 => ? = 3 - 1
[[1,4,5],[2,7],[3],[6]]
 => [6,3,2,7,1,4,5] => [6,3,2,7,1,5,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => 1 = 2 - 1
[[1,3,5],[2,7],[4],[6]]
 => [6,4,2,7,1,3,5] => [6,4,2,7,1,5,3] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
 => 1 = 2 - 1
[[1,3,6],[2,5],[4],[7]]
 => [7,4,2,5,1,3,6] => [7,4,2,6,1,5,3] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 3 - 1
[[1,3,5],[2,6],[4],[7]]
 => [7,4,2,6,1,3,5] => [7,4,2,6,1,5,3] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 3 - 1
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [7,4,3,2,1,6,5] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [7,5,3,2,1,6,4] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [7,5,4,2,1,6,3] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [7,6,3,2,1,5,4] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [7,6,4,2,1,5,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,4],[2,6],[3,7],[5]]
 => [5,3,7,2,6,1,4] => [5,3,7,2,6,1,4] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
 => 1 = 2 - 1
[[1,3],[2,6],[4,7],[5]]
 => [5,4,7,2,6,1,3] => [5,4,7,2,6,1,3] => ([(0,5),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6)],7)
 => 1 = 2 - 1
[[1,4],[2,5],[3,7],[6]]
 => [6,3,7,2,5,1,4] => [6,3,7,2,5,1,4] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => 1 = 2 - 1
[[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
 => 1 = 2 - 1
[[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [7,4,6,2,5,1,3] => ([(1,5),(2,5),(2,6),(3,4),(3,6)],7)
 => ? = 2 - 1
[[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [6,4,3,2,7,1,5] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [6,5,3,2,7,1,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [6,5,4,2,7,1,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,5],[2,6],[3],[4],[7]]
 => [7,4,3,2,6,1,5] => [7,4,3,2,6,1,5] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [7,5,3,2,6,1,4] => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [7,5,4,2,6,1,3] => ([(1,6),(2,6),(3,5),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,4],[2,5],[3],[6],[7]]
 => [7,6,3,2,5,1,4] => [7,6,3,2,5,1,4] => ([(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [7,6,4,2,5,1,3] => ([(2,6),(3,5),(4,5),(4,6)],7)
 => 1 = 2 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [7,5,4,3,2,1,6] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => ([(3,6),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ([(4,6),(5,6)],7)
 => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([],7)
 => 0 = 1 - 1

Description

The rank of the largest boolean interval in a poset.

Matching statistic: St000486
(load all 77 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1] => [1] => [1] => ? = 0 - 1
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [5,3,1,4,2] => [1,4,2,3,5] => 1 = 2 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [1,5,4,3,6,2] => [1,5,2,3,6,4] => 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [1,6,4,3,5,2] => [1,6,2,3,5,4] => 1 = 2 - 1
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [1,4,2,6,3,5] => 1 = 2 - 1
[[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [6,4,3,1,5,2] => [1,5,2,3,4,6] => 1 = 2 - 1
[[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [6,5,3,1,4,2] => [1,4,2,3,5,6] => 1 = 2 - 1
[[1,4],[2,5],[3],[6]]
 => [6,3,2,5,1,4] => [1,5,4,2,6,3] => [1,5,2,6,3,4] => 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [1,6,4,2,5,3] => [1,6,2,5,3,4] => 1 = 2 - 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 1 = 2 - 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [1,2,3,6,5,4] => [1,2,3,6,4,5] => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
 => [7,4,2,1,3,5,6] => [1,6,4,3,5,7,2] => [1,6,2,3,5,7,4] => ? = 3 - 1
[[1,3,6],[2,5],[4,7]]
 => [4,7,2,5,1,3,6] => [5,1,4,7,3,6,2] => [1,4,7,2,3,6,5] => ? = 3 - 1
[[1,4,5],[2,7],[3],[6]]
 => [6,3,2,7,1,4,5] => [7,4,3,1,5,6,2] => [1,5,6,2,3,4,7] => ? = 2 - 1
[[1,3,5],[2,7],[4],[6]]
 => [6,4,2,7,1,3,5] => [7,5,3,1,4,6,2] => [1,4,6,2,3,5,7] => 1 = 2 - 1
[[1,3,6],[2,5],[4],[7]]
 => [7,4,2,5,1,3,6] => [1,6,4,7,3,5,2] => [1,6,2,3,5,4,7] => ? = 3 - 1
[[1,3,5],[2,6],[4],[7]]
 => [7,4,2,6,1,3,5] => [1,6,4,2,5,7,3] => [1,6,2,5,7,3,4] => ? = 3 - 1
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [1,6,5,4,3,7,2] => [1,6,2,3,7,4,5] => ? = 2 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [1,7,5,4,3,6,2] => [1,7,2,3,6,4,5] => ? = 2 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [1,7,6,4,3,5,2] => [1,7,2,3,5,4,6] => ? = 2 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => 1 = 2 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [1,2,7,5,4,6,3] => [1,2,7,3,4,6,5] => 1 = 2 - 1
[[1,4],[2,6],[3,7],[5]]
 => [5,3,7,2,6,1,4] => [6,4,1,5,2,7,3] => [1,5,2,7,3,4,6] => ? = 2 - 1
[[1,3],[2,6],[4,7],[5]]
 => [5,4,7,2,6,1,3] => [6,5,1,4,2,7,3] => [1,4,2,7,3,5,6] => 1 = 2 - 1
[[1,4],[2,5],[3,7],[6]]
 => [6,3,7,2,5,1,4] => [7,4,1,5,2,6,3] => [1,5,2,6,3,4,7] => ? = 2 - 1
[[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [7,5,1,4,2,6,3] => [1,4,2,6,3,5,7] => 1 = 2 - 1
[[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [1,6,2,5,3,7,4] => [1,6,2,5,3,7,4] => ? = 2 - 1
[[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [7,5,4,3,1,6,2] => [1,6,2,3,4,5,7] => ? = 2 - 1
[[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [7,6,4,3,1,5,2] => [1,5,2,3,4,6,7] => ? = 2 - 1
[[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [7,6,5,3,1,4,2] => [1,4,2,3,5,6,7] => 1 = 2 - 1
[[1,5],[2,6],[3],[4],[7]]
 => [7,4,3,2,6,1,5] => [1,6,5,4,2,7,3] => [1,6,2,7,3,4,5] => ? = 2 - 1
[[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [1,7,5,4,2,6,3] => [1,7,2,6,3,4,5] => ? = 2 - 1
[[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [1,7,6,4,2,5,3] => [1,7,2,5,3,4,6] => ? = 2 - 1
[[1,4],[2,5],[3],[6],[7]]
 => [7,6,3,2,5,1,4] => [1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => 1 = 2 - 1
[[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => 1 = 2 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 2 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => 1 = 2 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => 1 = 2 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1

Description

The number of cycles of length at least 3 of a permutation.

Matching statistic: St000779
(load all 5 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1] => [1] => [1] => ? = 0 - 1
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [5,3,1,4,2] => [4,3,5,1,2] => 1 = 2 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,5,4,3,2] => [1,3,4,5,2] => 1 = 2 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [1,5,4,3,6,2] => [1,6,4,5,3,2] => 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [1,6,4,3,5,2] => [1,5,4,6,3,2] => 1 = 2 - 1
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [6,4,5,1,2,3] => 1 = 2 - 1
[[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [6,4,3,1,5,2] => [5,3,4,6,1,2] => 1 = 2 - 1
[[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [6,5,3,1,4,2] => [4,3,5,1,6,2] => 1 = 2 - 1
[[1,4],[2,5],[3],[6]]
 => [6,3,2,5,1,4] => [1,5,4,2,6,3] => [1,6,4,5,2,3] => 1 = 2 - 1
[[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [1,6,4,2,5,3] => [1,5,4,6,2,3] => 1 = 2 - 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,6,5,4,3,2] => [1,3,4,5,6,2] => 1 = 2 - 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [1,2,6,5,4,3] => [1,2,4,5,6,3] => 1 = 2 - 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
 => [7,4,2,1,3,5,6] => [1,6,4,3,5,7,2] => [1,7,4,6,3,5,2] => ? = 3 - 1
[[1,3,6],[2,5],[4,7]]
 => [4,7,2,5,1,3,6] => [5,1,4,7,3,6,2] => [6,7,5,1,4,3,2] => ? = 3 - 1
[[1,4,5],[2,7],[3],[6]]
 => [6,3,2,7,1,4,5] => [7,4,3,1,5,6,2] => [6,3,4,7,1,5,2] => ? = 2 - 1
[[1,3,5],[2,7],[4],[6]]
 => [6,4,2,7,1,3,5] => [7,5,3,1,4,6,2] => [6,3,5,1,7,4,2] => ? = 2 - 1
[[1,3,6],[2,5],[4],[7]]
 => [7,4,2,5,1,3,6] => [1,6,4,7,3,5,2] => [1,5,7,6,4,3,2] => ? = 3 - 1
[[1,3,5],[2,6],[4],[7]]
 => [7,4,2,6,1,3,5] => [1,6,4,2,5,7,3] => [1,7,4,6,2,5,3] => ? = 3 - 1
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [1,6,5,4,3,7,2] => [1,7,4,5,6,3,2] => ? = 2 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [1,7,5,4,3,6,2] => [1,6,4,5,7,3,2] => ? = 2 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [1,7,6,4,3,5,2] => [1,5,4,6,3,7,2] => ? = 2 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,2,6,5,4,7,3] => [1,2,7,5,6,4,3] => 1 = 2 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [1,2,7,5,4,6,3] => [1,2,6,5,7,4,3] => 1 = 2 - 1
[[1,4],[2,6],[3,7],[5]]
 => [5,3,7,2,6,1,4] => [6,4,1,5,2,7,3] => [7,5,4,6,1,2,3] => ? = 2 - 1
[[1,3],[2,6],[4,7],[5]]
 => [5,4,7,2,6,1,3] => [6,5,1,4,2,7,3] => [7,4,5,1,6,2,3] => ? = 2 - 1
[[1,4],[2,5],[3,7],[6]]
 => [6,3,7,2,5,1,4] => [7,4,1,5,2,6,3] => [6,5,4,7,1,2,3] => ? = 2 - 1
[[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [7,5,1,4,2,6,3] => [6,4,5,1,7,2,3] => ? = 2 - 1
[[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [1,6,2,5,3,7,4] => [1,7,5,6,2,3,4] => ? = 2 - 1
[[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [7,5,4,3,1,6,2] => [6,3,4,5,7,1,2] => ? = 2 - 1
[[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [7,6,4,3,1,5,2] => [5,3,4,6,1,7,2] => ? = 2 - 1
[[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [7,6,5,3,1,4,2] => [4,3,5,1,6,7,2] => ? = 2 - 1
[[1,5],[2,6],[3],[4],[7]]
 => [7,4,3,2,6,1,5] => [1,6,5,4,2,7,3] => [1,7,4,5,6,2,3] => ? = 2 - 1
[[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [1,7,5,4,2,6,3] => [1,6,4,5,7,2,3] => ? = 2 - 1
[[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [1,7,6,4,2,5,3] => [1,5,4,6,2,7,3] => ? = 2 - 1
[[1,4],[2,5],[3],[6],[7]]
 => [7,6,3,2,5,1,4] => [1,2,6,5,3,7,4] => [1,2,7,5,6,3,4] => 1 = 2 - 1
[[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [1,2,7,5,3,6,4] => [1,2,6,5,7,3,4] => 1 = 2 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => 1 = 2 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,2,7,6,5,4,3] => [1,2,4,5,6,7,3] => 1 = 2 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => 1 = 2 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => 1 = 2 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1

Description

The tier of a permutation. This is the number of elements

$i$ such that

$[i+1,k,i]$ is an occurrence of the pattern

$[2,3,1]$ . For example,

$[3,5,6,1,2,4]$ has tier

$2$ , with witnesses

$[3,5,2]$ (or

$[3,6,2]$ ) and

$[5,6,4]$ . According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].

Matching statistic: St000374
(load all 16 compositions to match this statistic)

Mapping

Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [[1]]
 => [1] => [1] => 0
[[1],[2]]
 => [[1],[2]]
 => [2,1] => [2,1] => 1
[[1],[2],[3]]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[[1,3],[2],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 2
[[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[2,5],[4]]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [5,3,4,1,2] => 2
[[1,4],[2],[3],[5]]
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [5,4,2,1,3] => 2
[[1,3],[2],[4],[5]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => 2
[[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
[[1,4,5],[2],[3],[6]]
 => [[1,3,6],[2],[4],[5]]
 => [5,4,2,1,3,6] => [5,4,2,1,6,3] => 2
[[1,3,5],[2],[4],[6]]
 => [[1,2,4],[3],[5],[6]]
 => [6,5,3,1,2,4] => [6,5,3,1,4,2] => 2
[[1,3],[2,5],[4,6]]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 2
[[1,4],[2,6],[3],[5]]
 => [[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [6,4,2,5,1,3] => 2
[[1,3],[2,6],[4],[5]]
 => [[1,2],[3,5],[4],[6]]
 => [6,4,3,5,1,2] => [6,4,3,5,1,2] => 2
[[1,4],[2,5],[3],[6]]
 => [[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [5,4,2,6,1,3] => 2
[[1,3],[2,5],[4],[6]]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [6,5,3,4,1,2] => 2
[[1,5],[2],[3],[4],[6]]
 => [[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [6,5,3,2,1,4] => 2
[[1,4],[2],[3],[5],[6]]
 => [[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [6,5,4,2,1,3] => 2
[[1,3],[2],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => 2
[[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[[1,3,5,6],[2],[4],[7]]
 => [[1,2,4,7],[3],[5],[6]]
 => [6,5,3,1,2,4,7] => [6,5,3,1,7,4,2] => ? = 3
[[1,3,6],[2,5],[4,7]]
 => [[1,2,4],[3,5],[6,7]]
 => [6,7,3,5,1,2,4] => [6,7,3,5,1,4,2] => ? = 3
[[1,4,5],[2,7],[3],[6]]
 => [[1,3,6],[2,7],[4],[5]]
 => [5,4,2,7,1,3,6] => [5,4,2,7,1,6,3] => ? = 2
[[1,3,5],[2,7],[4],[6]]
 => [[1,2,4],[3,6],[5],[7]]
 => [7,5,3,6,1,2,4] => [7,5,3,6,1,4,2] => ? = 2
[[1,3,6],[2,5],[4],[7]]
 => [[1,2,4],[3,5],[6],[7]]
 => [7,6,3,5,1,2,4] => [7,6,3,5,1,4,2] => ? = 3
[[1,3,5],[2,6],[4],[7]]
 => [[1,2,7],[3,4],[5],[6]]
 => [6,5,3,4,1,2,7] => [6,5,3,7,1,4,2] => ? = 3
[[1,5,6],[2],[3],[4],[7]]
 => [[1,4,7],[2],[3],[5],[6]]
 => [6,5,3,2,1,4,7] => [6,5,3,2,1,7,4] => ? = 2
[[1,4,6],[2],[3],[5],[7]]
 => [[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [7,6,4,2,1,5,3] => ? = 2
[[1,3,6],[2],[4],[5],[7]]
 => [[1,2,5],[3],[4],[6],[7]]
 => [7,6,4,3,1,2,5] => [7,6,4,3,1,5,2] => ? = 2
[[1,4,5],[2],[3],[6],[7]]
 => [[1,3,7],[2],[4],[5],[6]]
 => [6,5,4,2,1,3,7] => [6,5,4,2,1,7,3] => ? = 2
[[1,3,5],[2],[4],[6],[7]]
 => [[1,2,4],[3],[5],[6],[7]]
 => [7,6,5,3,1,2,4] => [7,6,5,3,1,4,2] => ? = 2
[[1,4],[2,6],[3,7],[5]]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => 2
[[1,3],[2,6],[4,7],[5]]
 => [[1,2],[3,5],[4,7],[6]]
 => [6,4,7,3,5,1,2] => [6,4,7,3,5,1,2] => 2
[[1,4],[2,5],[3,7],[6]]
 => [[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [7,4,6,2,5,1,3] => ? = 2
[[1,3],[2,5],[4,7],[6]]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => 2
[[1,3],[2,5],[4,6],[7]]
 => [[1,2],[3,4],[5,7],[6]]
 => [6,5,7,3,4,1,2] => [6,5,7,3,4,1,2] => 2
[[1,5],[2,7],[3],[4],[6]]
 => [[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [7,5,3,2,6,1,4] => ? = 2
[[1,4],[2,7],[3],[5],[6]]
 => [[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [7,5,4,2,6,1,3] => ? = 2
[[1,3],[2,7],[4],[5],[6]]
 => [[1,2],[3,6],[4],[5],[7]]
 => [7,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => ? = 2
[[1,5],[2,6],[3],[4],[7]]
 => [[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [6,5,3,2,7,1,4] => ? = 2
[[1,4],[2,6],[3],[5],[7]]
 => [[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [7,6,4,2,5,1,3] => ? = 2
[[1,3],[2,6],[4],[5],[7]]
 => [[1,2],[3,5],[4],[6],[7]]
 => [7,6,4,3,5,1,2] => [7,6,4,3,5,1,2] => ? = 2
[[1,4],[2,5],[3],[6],[7]]
 => [[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [6,5,4,2,7,1,3] => ? = 2
[[1,3],[2,5],[4],[6],[7]]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => 2
[[1,6],[2],[3],[4],[5],[7]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 2
[[1,5],[2],[3],[4],[6],[7]]
 => [[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => ? = 2
[[1,4],[2],[3],[5],[6],[7]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 2
[[1,3],[2],[4],[5],[6],[7]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => 2
[[1],[2],[3],[4],[5],[6],[7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 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: St000485
(load all 5 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1] => [1] => [1] => ? = 0
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => [3,5,1,4,2] => 2
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [1,4,5,6,3,2] => [1,6,5,4,3,2] => 2
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => 2
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => 2
[[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [2,4,5,1,6,3] => [4,6,3,1,5,2] => 2
[[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => 2
[[1,4],[2,5],[3],[6]]
 => [6,3,2,5,1,4] => [1,4,5,2,6,3] => [1,6,4,3,5,2] => 2
[[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => 2
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 2
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 2
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[[1,3,5,6],[2],[4],[7]]
 => [7,4,2,1,3,5,6] => [1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => ? = 3
[[1,3,6],[2,5],[4,7]]
 => [4,7,2,5,1,3,6] => [4,1,6,3,7,5,2] => [7,2,6,4,5,3,1] => ? = 3
[[1,4,5],[2,7],[3],[6]]
 => [6,3,2,7,1,4,5] => [2,5,6,1,7,4,3] => [4,7,6,1,5,3,2] => ? = 2
[[1,3,5],[2,7],[4],[6]]
 => [6,4,2,7,1,3,5] => [2,4,6,1,7,5,3] => [4,7,6,1,5,3,2] => ? = 2
[[1,3,6],[2,5],[4],[7]]
 => [7,4,2,5,1,3,6] => [1,4,6,3,7,5,2] => [1,7,6,4,5,3,2] => ? = 3
[[1,3,5],[2,6],[4],[7]]
 => [7,4,2,6,1,3,5] => [1,4,6,2,7,5,3] => [1,7,6,4,5,3,2] => ? = 3
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [1,4,5,6,7,3,2] => [1,7,6,4,5,3,2] => ? = 2
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => ? = 2
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => ? = 2
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,2,5,6,7,4,3] => [1,2,7,6,5,4,3] => 2
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => 2
[[1,4],[2,6],[3,7],[5]]
 => [5,3,7,2,6,1,4] => [3,5,1,6,2,7,4] => [5,7,3,4,1,6,2] => ? = 2
[[1,3],[2,6],[4,7],[5]]
 => [5,4,7,2,6,1,3] => [3,4,1,6,2,7,5] => [5,3,2,7,1,6,4] => ? = 2
[[1,4],[2,5],[3,7],[6]]
 => [6,3,7,2,5,1,4] => [2,5,1,6,3,7,4] => [3,7,1,5,4,6,2] => ? = 2
[[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [2,4,1,6,3,7,5] => [3,5,1,7,2,6,4] => ? = 2
[[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [1,4,2,6,3,7,5] => [1,5,3,7,2,6,4] => ? = 2
[[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [2,4,5,6,1,7,3] => [5,7,3,4,1,6,2] => ? = 2
[[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ? = 2
[[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [2,3,4,6,1,7,5] => [5,2,3,7,1,6,4] => ? = 2
[[1,5],[2,6],[3],[4],[7]]
 => [7,4,3,2,6,1,5] => [1,4,5,6,2,7,3] => [1,7,5,4,3,6,2] => ? = 2
[[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [1,3,5,6,2,7,4] => [1,5,7,4,2,6,3] => ? = 2
[[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [1,3,4,6,2,7,5] => [1,5,3,7,2,6,4] => ? = 2
[[1,4],[2,5],[3],[6],[7]]
 => [7,6,3,2,5,1,4] => [1,2,5,6,3,7,4] => [1,2,7,5,4,6,3] => 2
[[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [1,2,4,6,3,7,5] => [1,2,5,7,3,6,4] => 2
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 2
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => 2
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => 2
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 2
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1

Description

The length of the longest cycle of a permutation.

Matching statistic: St000834
(load all 2 compositions to match this statistic)

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [[1]]
 => [1] => [1] => 0
[[1],[2]]
 => [[1,2]]
 => [1,2] => [1,2] => 1
[[1],[2],[3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => 2
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 1
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,4,5,1,3] => 2
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => 2
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => 2
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[[1,4,5],[2],[3],[6]]
 => [[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [2,3,5,4,1,6] => 2
[[1,3,5],[2],[4],[6]]
 => [[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => [2,4,5,3,1,6] => 2
[[1,3],[2,5],[4,6]]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => [2,4,6,1,3,5] => 2
[[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [2,3,5,6,1,4] => 2
[[1,3],[2,6],[4],[5]]
 => [[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [2,4,5,6,1,3] => 2
[[1,4],[2,5],[3],[6]]
 => [[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [2,3,5,1,4,6] => 2
[[1,3],[2,5],[4],[6]]
 => [[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [2,4,5,1,3,6] => 2
[[1,5],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 2
[[1,4],[2],[3],[5],[6]]
 => [[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [2,3,4,1,5,6] => 2
[[1,3],[2],[4],[5],[6]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [2,3,1,4,5,6] => 2
[[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[[1,3,5,6],[2],[4],[7]]
 => [[1,2,4,7],[3],[5],[6]]
 => [6,5,3,1,2,4,7] => [2,4,6,5,3,1,7] => ? = 3
[[1,3,6],[2,5],[4,7]]
 => [[1,2,4],[3,5,7],[6]]
 => [6,3,5,7,1,2,4] => [2,4,7,1,5,6,3] => ? = 3
[[1,4,5],[2,7],[3],[6]]
 => [[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [2,3,6,7,1,5,4] => ? = 2
[[1,3,5],[2,7],[4],[6]]
 => [[1,2,4,6],[3,7],[5]]
 => [5,3,7,1,2,4,6] => [2,4,6,7,1,5,3] => ? = 2
[[1,3,6],[2,5],[4],[7]]
 => [[1,2,4,7],[3,5],[6]]
 => [6,3,5,1,2,4,7] => [2,4,5,1,6,3,7] => ? = 3
[[1,3,5],[2,6],[4],[7]]
 => [[1,2,4,7],[3,6],[5]]
 => [5,3,6,1,2,4,7] => [2,4,6,1,5,3,7] => ? = 3
[[1,5,6],[2],[3],[4],[7]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [2,3,4,6,5,1,7] => ? = 2
[[1,4,6],[2],[3],[5],[7]]
 => [[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [2,3,5,6,4,1,7] => ? = 2
[[1,3,6],[2],[4],[5],[7]]
 => [[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [2,4,5,6,3,1,7] => ? = 2
[[1,4,5],[2],[3],[6],[7]]
 => [[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [2,3,5,4,1,6,7] => ? = 2
[[1,3,5],[2],[4],[6],[7]]
 => [[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [2,4,5,3,1,6,7] => ? = 2
[[1,4],[2,6],[3,7],[5]]
 => [[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2
[[1,3],[2,6],[4,7],[5]]
 => [[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [2,4,5,7,1,3,6] => ? = 2
[[1,4],[2,5],[3,7],[6]]
 => [[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ? = 2
[[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => [2,4,6,7,1,3,5] => ? = 2
[[1,3],[2,5],[4,6],[7]]
 => [[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => [2,4,6,1,3,5,7] => ? = 2
[[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => 2
[[1,4],[2,7],[3],[5],[6]]
 => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [2,3,5,6,7,1,4] => 2
[[1,3],[2,7],[4],[5],[6]]
 => [[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [2,4,5,6,7,1,3] => ? = 2
[[1,5],[2,6],[3],[4],[7]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [2,3,4,6,1,5,7] => ? = 2
[[1,4],[2,6],[3],[5],[7]]
 => [[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => [2,3,5,6,1,4,7] => ? = 2
[[1,3],[2,6],[4],[5],[7]]
 => [[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => [2,4,5,6,1,3,7] => ? = 2
[[1,4],[2,5],[3],[6],[7]]
 => [[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [2,3,5,1,4,6,7] => ? = 2
[[1,3],[2,5],[4],[6],[7]]
 => [[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [2,4,5,1,3,6,7] => ? = 2
[[1,6],[2],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => 2
[[1,5],[2],[3],[4],[6],[7]]
 => [[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => 2
[[1,4],[2],[3],[5],[6],[7]]
 => [[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => 2
[[1,3],[2],[4],[5],[6],[7]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => 2
[[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 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

$[w_1,..., w_n,0]$ .

Matching statistic: St000451
(load all 17 compositions to match this statistic)

Mapping

Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [[1]]
 => [1] => [1] => 1 = 0 + 1
[[1],[2]]
 => [[1],[2]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[2],[3]]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [5,3,4,1,2] => 3 = 2 + 1
[[1,4],[2],[3],[5]]
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [5,4,2,1,3] => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => 3 = 2 + 1
[[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 2 = 1 + 1
[[1,4,5],[2],[3],[6]]
 => [[1,3,6],[2],[4],[5]]
 => [5,4,2,1,3,6] => [5,4,2,1,6,3] => 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
 => [[1,2,4],[3],[5],[6]]
 => [6,5,3,1,2,4] => [6,5,3,1,4,2] => 3 = 2 + 1
[[1,3],[2,5],[4,6]]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
 => [[1,3],[2,5],[4],[6]]
 => [6,4,2,5,1,3] => [6,4,2,5,1,3] => 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
 => [[1,2],[3,5],[4],[6]]
 => [6,4,3,5,1,2] => [6,4,3,5,1,2] => 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
 => [[1,3],[2,6],[4],[5]]
 => [5,4,2,6,1,3] => [5,4,2,6,1,3] => 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [6,5,3,4,1,2] => 3 = 2 + 1
[[1,5],[2],[3],[4],[6]]
 => [[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [6,5,3,2,1,4] => 3 = 2 + 1
[[1,4],[2],[3],[5],[6]]
 => [[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [6,5,4,2,1,3] => 3 = 2 + 1
[[1,3],[2],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => 3 = 2 + 1
[[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 2 = 1 + 1
[[1,3,5,6],[2],[4],[7]]
 => [[1,2,4,7],[3],[5],[6]]
 => [6,5,3,1,2,4,7] => [6,5,3,1,7,4,2] => ? = 3 + 1
[[1,3,6],[2,5],[4,7]]
 => [[1,2,4],[3,5],[6,7]]
 => [6,7,3,5,1,2,4] => [6,7,3,5,1,4,2] => ? = 3 + 1
[[1,4,5],[2,7],[3],[6]]
 => [[1,3,6],[2,7],[4],[5]]
 => [5,4,2,7,1,3,6] => [5,4,2,7,1,6,3] => ? = 2 + 1
[[1,3,5],[2,7],[4],[6]]
 => [[1,2,4],[3,6],[5],[7]]
 => [7,5,3,6,1,2,4] => [7,5,3,6,1,4,2] => ? = 2 + 1
[[1,3,6],[2,5],[4],[7]]
 => [[1,2,4],[3,5],[6],[7]]
 => [7,6,3,5,1,2,4] => [7,6,3,5,1,4,2] => ? = 3 + 1
[[1,3,5],[2,6],[4],[7]]
 => [[1,2,7],[3,4],[5],[6]]
 => [6,5,3,4,1,2,7] => [6,5,3,7,1,4,2] => ? = 3 + 1
[[1,5,6],[2],[3],[4],[7]]
 => [[1,4,7],[2],[3],[5],[6]]
 => [6,5,3,2,1,4,7] => [6,5,3,2,1,7,4] => ? = 2 + 1
[[1,4,6],[2],[3],[5],[7]]
 => [[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [7,6,4,2,1,5,3] => ? = 2 + 1
[[1,3,6],[2],[4],[5],[7]]
 => [[1,2,5],[3],[4],[6],[7]]
 => [7,6,4,3,1,2,5] => [7,6,4,3,1,5,2] => ? = 2 + 1
[[1,4,5],[2],[3],[6],[7]]
 => [[1,3,7],[2],[4],[5],[6]]
 => [6,5,4,2,1,3,7] => [6,5,4,2,1,7,3] => ? = 2 + 1
[[1,3,5],[2],[4],[6],[7]]
 => [[1,2,4],[3],[5],[6],[7]]
 => [7,6,5,3,1,2,4] => [7,6,5,3,1,4,2] => ? = 2 + 1
[[1,4],[2,6],[3,7],[5]]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => 3 = 2 + 1
[[1,3],[2,6],[4,7],[5]]
 => [[1,2],[3,5],[4,7],[6]]
 => [6,4,7,3,5,1,2] => [6,4,7,3,5,1,2] => 3 = 2 + 1
[[1,4],[2,5],[3,7],[6]]
 => [[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [7,4,6,2,5,1,3] => ? = 2 + 1
[[1,3],[2,5],[4,7],[6]]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => 3 = 2 + 1
[[1,3],[2,5],[4,6],[7]]
 => [[1,2],[3,4],[5,7],[6]]
 => [6,5,7,3,4,1,2] => [6,5,7,3,4,1,2] => 3 = 2 + 1
[[1,5],[2,7],[3],[4],[6]]
 => [[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [7,5,3,2,6,1,4] => ? = 2 + 1
[[1,4],[2,7],[3],[5],[6]]
 => [[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [7,5,4,2,6,1,3] => ? = 2 + 1
[[1,3],[2,7],[4],[5],[6]]
 => [[1,2],[3,6],[4],[5],[7]]
 => [7,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => ? = 2 + 1
[[1,5],[2,6],[3],[4],[7]]
 => [[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [6,5,3,2,7,1,4] => ? = 2 + 1
[[1,4],[2,6],[3],[5],[7]]
 => [[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [7,6,4,2,5,1,3] => ? = 2 + 1
[[1,3],[2,6],[4],[5],[7]]
 => [[1,2],[3,5],[4],[6],[7]]
 => [7,6,4,3,5,1,2] => [7,6,4,3,5,1,2] => ? = 2 + 1
[[1,4],[2,5],[3],[6],[7]]
 => [[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [6,5,4,2,7,1,3] => ? = 2 + 1
[[1,3],[2,5],[4],[6],[7]]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => 3 = 2 + 1
[[1,6],[2],[3],[4],[5],[7]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 2 + 1
[[1,5],[2],[3],[4],[6],[7]]
 => [[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => ? = 2 + 1
[[1,4],[2],[3],[5],[6],[7]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 2 + 1
[[1,3],[2],[4],[5],[6],[7]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => 3 = 2 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 2 = 1 + 1

Description

The length of the longest pattern of the form k 1 2...(k-1).

Matching statistic: St000891
(load all 4 compositions to match this statistic)

Mapping

Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => {{1}}
 => [1] => [1] => 1 = 0 + 1
[[1],[2]]
 => {{1},{2}}
 => [1,2] => [1,2] => 2 = 1 + 1
[[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 2 = 1 + 1
[[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,5,3,4,2] => 3 = 2 + 1
[[1,4],[2],[3],[5]]
 => {{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [2,4,3,1,5] => 3 = 2 + 1
[[1,3],[2],[4],[5]]
 => {{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 1 + 1
[[1,4,5],[2],[3],[6]]
 => {{1,4,5},{2},{3},{6}}
 => [4,2,3,5,1,6] => [2,3,1,5,4,6] => 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
 => {{1,3,5},{2},{4},{6}}
 => [3,2,5,4,1,6] => [1,5,2,4,3,6] => 3 = 2 + 1
[[1,3],[2,5],[4,6]]
 => {{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => [1,5,6,3,2,4] => 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => [1,6,4,3,5,2] => 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
 => {{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [1,4,6,3,5,2] => 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
 => {{1,4},{2,5},{3},{6}}
 => [4,5,3,1,2,6] => [4,1,5,3,2,6] => 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
 => {{1,3},{2,5},{4},{6}}
 => [3,5,1,4,2,6] => [1,5,3,4,2,6] => 3 = 2 + 1
[[1,5],[2],[3],[4],[6]]
 => {{1,5},{2},{3},{4},{6}}
 => [5,2,3,4,1,6] => [2,3,5,4,1,6] => 3 = 2 + 1
[[1,4],[2],[3],[5],[6]]
 => {{1,4},{2},{3},{5},{6}}
 => [4,2,3,1,5,6] => [2,4,3,1,5,6] => 3 = 2 + 1
[[1,3],[2],[4],[5],[6]]
 => {{1,3},{2},{4},{5},{6}}
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 3 = 2 + 1
[[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 1 + 1
[[1,3,5,6],[2],[4],[7]]
 => {{1,3,5,6},{2},{4},{7}}
 => [3,2,5,4,6,1,7] => [1,4,2,3,6,5,7] => 4 = 3 + 1
[[1,3,6],[2,5],[4,7]]
 => {{1,3,6},{2,5},{4,7}}
 => [3,5,6,7,2,1,4] => [2,6,1,5,3,7,4] => ? = 3 + 1
[[1,4,5],[2,7],[3],[6]]
 => {{1,4,5},{2,7},{3},{6}}
 => [4,7,3,5,1,6,2] => [1,3,7,5,4,6,2] => 3 = 2 + 1
[[1,3,5],[2,7],[4],[6]]
 => {{1,3,5},{2,7},{4},{6}}
 => [3,7,5,4,1,6,2] => [7,5,1,4,3,6,2] => ? = 2 + 1
[[1,3,6],[2,5],[4],[7]]
 => {{1,3,6},{2,5},{4},{7}}
 => [3,5,6,4,2,1,7] => [2,6,1,5,4,3,7] => ? = 3 + 1
[[1,3,5],[2,6],[4],[7]]
 => {{1,3,5},{2,6},{4},{7}}
 => [3,6,5,4,1,2,7] => [5,6,1,4,3,2,7] => ? = 3 + 1
[[1,5,6],[2],[3],[4],[7]]
 => {{1,5,6},{2},{3},{4},{7}}
 => [5,2,3,4,6,1,7] => [2,3,4,1,6,5,7] => ? = 2 + 1
[[1,4,6],[2],[3],[5],[7]]
 => {{1,4,6},{2},{3},{5},{7}}
 => [4,2,3,6,5,1,7] => [6,2,1,3,5,4,7] => ? = 2 + 1
[[1,3,6],[2],[4],[5],[7]]
 => {{1,3,6},{2},{4},{5},{7}}
 => [3,2,6,4,5,1,7] => [1,4,6,2,5,3,7] => 3 = 2 + 1
[[1,4,5],[2],[3],[6],[7]]
 => {{1,4,5},{2},{3},{6},{7}}
 => [4,2,3,5,1,6,7] => [2,3,1,5,4,6,7] => ? = 2 + 1
[[1,3,5],[2],[4],[6],[7]]
 => {{1,3,5},{2},{4},{6},{7}}
 => [3,2,5,4,1,6,7] => [1,5,2,4,3,6,7] => ? = 2 + 1
[[1,4],[2,6],[3,7],[5]]
 => {{1,4},{2,6},{3,7},{5}}
 => [4,6,7,1,5,2,3] => [1,6,7,4,2,5,3] => ? = 2 + 1
[[1,3],[2,6],[4,7],[5]]
 => {{1,3},{2,6},{4,7},{5}}
 => [3,6,1,7,5,2,4] => [6,7,1,5,3,2,4] => ? = 2 + 1
[[1,4],[2,5],[3,7],[6]]
 => {{1,4},{2,5},{3,7},{6}}
 => [4,5,7,1,2,6,3] => [1,7,4,2,5,6,3] => ? = 2 + 1
[[1,3],[2,5],[4,7],[6]]
 => {{1,3},{2,5},{4,7},{6}}
 => [3,5,1,7,2,6,4] => [7,1,5,2,6,3,4] => ? = 2 + 1
[[1,3],[2,5],[4,6],[7]]
 => {{1,3},{2,5},{4,6},{7}}
 => [3,5,1,6,2,4,7] => [1,5,6,3,2,4,7] => ? = 2 + 1
[[1,5],[2,7],[3],[4],[6]]
 => {{1,5},{2,7},{3},{4},{6}}
 => [5,7,3,4,1,6,2] => [3,1,7,5,4,6,2] => ? = 2 + 1
[[1,4],[2,7],[3],[5],[6]]
 => {{1,4},{2,7},{3},{5},{6}}
 => [4,7,3,1,5,6,2] => [1,5,3,7,4,6,2] => ? = 2 + 1
[[1,3],[2,7],[4],[5],[6]]
 => {{1,3},{2,7},{4},{5},{6}}
 => [3,7,1,4,5,6,2] => [1,4,5,7,3,6,2] => 3 = 2 + 1
[[1,5],[2,6],[3],[4],[7]]
 => {{1,5},{2,6},{3},{4},{7}}
 => [5,6,3,4,1,2,7] => [5,3,1,6,4,2,7] => ? = 2 + 1
[[1,4],[2,6],[3],[5],[7]]
 => {{1,4},{2,6},{3},{5},{7}}
 => [4,6,3,1,5,2,7] => [1,6,4,3,5,2,7] => ? = 2 + 1
[[1,3],[2,6],[4],[5],[7]]
 => {{1,3},{2,6},{4},{5},{7}}
 => [3,6,1,4,5,2,7] => [1,4,6,3,5,2,7] => 3 = 2 + 1
[[1,4],[2,5],[3],[6],[7]]
 => {{1,4},{2,5},{3},{6},{7}}
 => [4,5,3,1,2,6,7] => [4,1,5,3,2,6,7] => ? = 2 + 1
[[1,3],[2,5],[4],[6],[7]]
 => {{1,3},{2,5},{4},{6},{7}}
 => [3,5,1,4,2,6,7] => [1,5,3,4,2,6,7] => ? = 2 + 1
[[1,6],[2],[3],[4],[5],[7]]
 => {{1,6},{2},{3},{4},{5},{7}}
 => [6,2,3,4,5,1,7] => [2,3,4,6,5,1,7] => ? = 2 + 1
[[1,5],[2],[3],[4],[6],[7]]
 => {{1,5},{2},{3},{4},{6},{7}}
 => [5,2,3,4,1,6,7] => [2,3,5,4,1,6,7] => ? = 2 + 1
[[1,4],[2],[3],[5],[6],[7]]
 => {{1,4},{2},{3},{5},{6},{7}}
 => [4,2,3,1,5,6,7] => [2,4,3,1,5,6,7] => ? = 2 + 1
[[1,3],[2],[4],[5],[6],[7]]
 => {{1,3},{2},{4},{5},{6},{7}}
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 3 = 2 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 2 = 1 + 1

Description

The number of distinct diagonal sums of a permutation matrix. For example, the sums of the diagonals of the matrix

$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$ are

$(1,0,1,0,2,0)$ , so the statistic is

$3$ .

The following 102 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000035The number of left outer peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001096The size of the overlap set of a permutation. St000353The number of inner valleys of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000245The number of ascents of a permutation. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St000864The number of circled entries of the shifted recording tableau of a permutation. St000990The first ascent of a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001737The number of descents of type 2 in a permutation. St000007The number of saliances of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000836The number of descents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001642The Prague dimension of a graph. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000243The number of cyclic valleys and cyclic peaks of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000654The first descent of a permutation. St000472The sum of the ascent bottoms of a permutation. St000646The number of big ascents of a permutation. St000732The number of double deficiencies of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001488The number of corners of a skew partition. St000711The number of big exceedences of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001469The holeyness of a permutation. St000092The number of outer peaks of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001571The Cartan determinant of the integer partition. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000264The girth of a graph, which is not a tree. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001209The pmaj statistic of a parking function. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001555The order of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000907The number of maximal antichains of minimal length in a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001927Sparre Andersen's number of positives of a signed permutation. St001498The normalised height of a Nakayama algebra with magnitude 1.