- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000352
(load all 6 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[(1,2)]
 => [1,0]
 => [1] => 0
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,2] => 0
[(1,3),(2,4)]
 => [1,1,0,0]
 => [2,1] => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [2,1] => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1

Description

The Elizalde-Pak rank of a permutation. This is the largest

$k$ such that

$\pi(i) > k$ for all

$i\leq k$ . According to [1], the length of the longest increasing subsequence in a

$321$ -avoiding permutation is equidistributed with the rank of a

$132$ -avoiding permutation.

Matching statistic: St000701
(load all 3 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%

Values

[(1,2)]
 => [2,1] => [1,2] => [.,[.,.]]
 => 1 = 0 + 1
[(1,2),(3,4)]
 => [2,1,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 1 = 0 + 1
[(1,3),(2,4)]
 => [3,4,1,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[(1,4),(2,3)]
 => [3,4,2,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,4,3,6,5,2] => [.,[[[.,.],[[.,.],.]],.]]
 => 1 = 0 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [4,1,2,6,5,3] => [[.,.],[.,[[[.,.],.],.]]]
 => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [3,1,2,6,5,4] => [[.,.],[.,[[[.,.],.],.]]]
 => 2 = 1 + 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [3,1,6,2,4,5] => [[.,.],[[.,.],[.,[.,.]]]]
 => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [3,1,5,2,4,6] => [[.,.],[[.,.],[.,[.,.]]]]
 => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [4,6,1,2,3,5] => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [5,1,6,2,4,3] => [[.,.],[[.,.],[[.,.],.]]]
 => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,5,6,3,4,2] => [.,[[[.,[.,.]],[.,.]],.]]
 => 1 = 0 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [1,6,5,3,4,2] => [.,[[[[.,.],.],[.,.]],.]]
 => 1 = 0 + 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [6,1,5,2,4,3] => [[.,.],[[.,.],[[.,.],.]]]
 => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [6,5,1,2,3,4] => [[[.,.],.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [6,4,1,2,3,5] => [[[.,.],.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [5,4,1,2,3,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,4,3,6,5,8,7,2] => [.,[[[.,.],[[.,.],[[.,.],.]]],.]]
 => 1 = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [4,1,2,6,5,8,7,3] => [[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [3,1,2,6,5,8,7,4] => [[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
 => [3,5,2,6,1,4,8,7] => [3,1,6,2,4,8,7,5] => [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => 2 = 1 + 1
[(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [3,1,5,2,4,8,7,6] => [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => 2 = 1 + 1
[(1,7),(2,3),(4,5),(6,8)]
 => [3,5,2,7,4,8,1,6] => [3,1,5,2,8,4,6,7] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => 2 = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [3,1,5,2,7,4,6,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
 => [4,5,7,2,3,8,6,1] => [4,5,1,2,7,3,6,8] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
 => 2 = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [4,5,1,2,8,3,6,7] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
 => 2 = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
 => [4,5,6,2,3,1,8,7] => [4,5,1,2,3,8,7,6] => [[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => 2 = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
 => [4,5,6,2,1,3,8,7] => [4,6,1,2,3,8,7,5] => [[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => 2 = 1 + 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [5,6,1,2,3,8,7,4] => [[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [5,1,6,2,4,8,7,3] => [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
 => 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,5,6,3,4,8,7,2] => [.,[[[.,[.,.]],[.,[[.,.],.]]],.]]
 => 1 = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [1,6,5,3,4,8,7,2] => [.,[[[[.,.],.],[.,[[.,.],.]]],.]]
 => 1 = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
 => [3,5,1,6,4,2,8,7] => [6,1,5,2,4,8,7,3] => [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
 => 2 = 1 + 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,5,6,1,3,2,8,7] => [6,5,1,2,3,8,7,4] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => ? = 1 + 1
[(1,5),(2,6),(3,4),(7,8)]
 => [4,5,6,3,1,2,8,7] => [6,4,1,2,3,8,7,5] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => ? = 1 + 1
[(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [5,4,1,2,3,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [5,4,1,2,8,3,6,7] => [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
 => ? = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [5,4,1,2,7,3,6,8] => [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
 => ? = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [6,4,1,7,2,3,5,8] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [6,4,1,8,2,3,5,7] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [7,4,1,8,2,3,5,6] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
 => [4,5,7,3,1,8,2,6] => [7,4,1,2,8,3,6,5] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
 => ? = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,5,7,1,3,8,2,6] => [7,5,1,2,8,3,6,4] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
 => ? = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,5,1,7,4,8,2,6] => [7,1,5,2,8,4,6,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
 => 2 = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,5,7,4,8,3,6] => [1,7,5,3,8,4,6,2] => [.,[[[[.,.],.],[[.,.],[.,.]]],.]]
 => 1 = 0 + 1
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [1,8,5,3,7,4,6,2] => [.,[[[[.,.],.],[[.,.],[.,.]]],.]]
 => 1 = 0 + 1
[(1,3),(2,8),(4,5),(6,7)]
 => [3,5,1,7,4,8,6,2] => [8,1,5,2,7,4,6,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
 => 2 = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,5,7,1,3,8,6,2] => [8,5,1,2,7,3,6,4] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
 => ? = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
 => [4,5,7,3,1,8,6,2] => [8,4,1,2,7,3,6,5] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
 => ? = 1 + 1
[(1,6),(2,8),(3,4),(5,7)]
 => [4,6,7,3,8,1,5,2] => [8,4,1,7,2,3,5,6] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 1 + 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [8,4,1,6,2,3,5,7] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 1 + 1
[(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [7,4,1,6,2,3,5,8] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [7,5,6,1,2,3,4,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [8,5,6,1,2,3,4,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [8,5,7,1,2,3,4,6] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,5),(2,8),(3,6),(4,7)]
 => [5,6,7,8,1,3,4,2] => [8,6,7,1,2,3,4,5] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,4),(2,8),(3,6),(5,7)]
 => [4,6,7,1,8,3,5,2] => [8,6,1,7,2,3,5,4] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
 => ? = 1 + 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,6,1,7,8,4,5,2] => [8,1,6,7,2,4,5,3] => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
 => 2 = 1 + 1
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,6,7,8,4,5,3] => [1,8,6,7,3,4,5,2] => [.,[[[[.,.],[.,.]],[.,[.,.]]],.]]
 => 1 = 0 + 1
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,6,7,8,4,3,5] => [1,7,6,8,3,4,5,2] => [.,[[[[.,.],[.,.]],[.,[.,.]]],.]]
 => 1 = 0 + 1
[(1,3),(2,7),(4,6),(5,8)]
 => [3,6,1,7,8,4,2,5] => [7,1,6,8,2,4,5,3] => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
 => 2 = 1 + 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,6,7,1,8,3,2,5] => [7,6,1,8,2,3,5,4] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
 => ? = 1 + 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,6,7,8,1,3,2,4] => [7,6,8,1,2,3,4,5] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [7,5,8,1,2,3,4,6] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [6,5,8,1,2,3,4,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [6,5,7,1,2,3,4,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [5,6,8,1,2,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [5,7,8,1,2,3,4,6] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [6,7,1,8,2,3,5,4] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
 => 2 = 1 + 1
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [6,1,7,8,2,4,5,3] => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
 => 2 = 1 + 1
[(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => [1,6,7,8,3,4,5,2] => [.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
 => 1 = 0 + 1
[(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [1,5,7,3,8,4,6,2] => [.,[[[.,[.,.]],[[.,.],[.,.]]],.]]
 => 1 = 0 + 1
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [5,1,7,2,8,4,6,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
 => 2 = 1 + 1
[(1,4),(2,5),(3,7),(6,8)]
 => [4,5,7,1,2,8,3,6] => [5,7,1,2,8,3,6,4] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
 => 2 = 1 + 1
[(1,5),(2,4),(3,7),(6,8)]
 => [4,5,7,2,1,8,3,6] => [4,7,1,2,8,3,6,5] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
 => 2 = 1 + 1
[(1,5),(2,6),(3,8),(4,7)]
 => [5,6,7,8,1,2,4,3] => [6,8,7,1,2,3,4,5] => [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,6),(2,5),(3,8),(4,7)]
 => [5,6,7,8,2,1,4,3] => [5,8,7,1,2,3,4,6] => [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [5,8,6,1,2,3,4,7] => [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,8),(2,5),(3,7),(4,6)]
 => [5,6,7,8,2,4,3,1] => [5,7,6,1,2,3,4,8] => [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [6,7,5,1,2,3,4,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [6,8,5,1,2,3,4,7] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,6),(2,7),(3,8),(4,5)]
 => [5,6,7,8,4,1,2,3] => [7,8,5,1,2,3,4,6] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,5),(2,7),(3,8),(4,6)]
 => [5,6,7,8,1,4,2,3] => [7,8,6,1,2,3,4,5] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,4),(2,8),(3,7),(5,6)]
 => [4,6,7,1,8,5,3,2] => [8,7,1,6,2,3,5,4] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
 => ? = 1 + 1
[(1,5),(2,8),(3,7),(4,6)]
 => [5,6,7,8,1,4,3,2] => [8,7,6,1,2,3,4,5] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,6),(2,8),(3,7),(4,5)]
 => [5,6,7,8,4,1,3,2] => [8,7,5,1,2,3,4,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,7),(2,8),(3,6),(4,5)]
 => [5,6,7,8,4,3,1,2] => [8,6,5,1,2,3,4,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [7,6,5,1,2,3,4,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,4,3,6,5,8,7,10,9,2] => [.,[[[.,.],[[.,.],[[.,.],[[.,.],.]]]],.]]
 => ? = 0 + 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => [4,1,2,6,5,8,7,10,9,3] => ?
 => ? = 1 + 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [3,1,2,6,5,8,7,10,9,4] => [[.,.],[.,[[[.,.],[[.,.],[[.,.],.]]],.]]]
 => ? = 1 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [3,5,2,6,1,4,8,7,10,9] => ? => ?
 => ? = 1 + 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [3,1,5,2,4,8,7,10,9,6] => [[.,.],[[.,.],[.,[[[.,.],[[.,.],.]],.]]]]
 => ? = 1 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [3,5,2,7,4,8,1,6,10,9] => ? => ?
 => ? = 1 + 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [3,5,2,7,4,8,6,1,10,9] => [3,1,5,2,7,4,6,10,9,8] => [[.,.],[[.,.],[[.,.],[.,[[[.,.],.],.]]]]]
 => ? = 1 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [3,5,2,7,4,9,6,10,1,8] => ? => ?
 => ? = 1 + 1

Description

The protection number of a binary tree. This is the minimal distance from the root to a leaf.

Matching statistic: St001085

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%

Values

[(1,2)]
 => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 1
[(1,4),(2,3)]
 => [3,4,2,1] => [1,3,2,4] => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 0
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [1,4,3,6,2,5] => [5,3,2,6,4,1] => 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [1,4,3,6,2,5] => [5,3,2,6,4,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 1
[(1,5),(2,3),(4,6),(7,8)]
 => [3,5,2,6,1,4,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 1
[(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 1
[(1,7),(2,3),(4,5),(6,8)]
 => [3,5,2,7,4,8,1,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [4,5,7,2,3,8,6,1] => [1,4,2,5,3,7,6,8] => [3,5,2,4,7,6,8,1] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [1,4,2,5,3,7,6,8] => [3,5,2,4,7,6,8,1] => ? = 1
[(1,6),(2,4),(3,5),(7,8)]
 => [4,5,6,2,3,1,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
 => [4,5,6,2,1,3,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [2,3,5,4,6,7,8,1] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [1,2,3,5,4,6,7,8] => [2,3,5,4,6,7,8,1] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,5,1,6,4,2,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,5,6,1,3,2,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [4,5,6,3,1,2,8,7] => [1,4,3,6,2,5,7,8] => [5,3,2,6,4,7,8,1] => ? = 1
[(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [1,4,3,6,2,5,7,8] => [5,3,2,6,4,7,8,1] => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [1,4,3,7,2,5,6,8] => [5,3,2,6,7,4,8,1] => ? = 1
[(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [1,4,3,7,6,8,2,5] => [7,3,2,8,5,4,6,1] => ? = 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [1,4,3,7,5,8,2,6] => [7,3,2,5,8,4,6,1] => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [1,4,3,7,2,6,5,8] => [5,3,2,7,6,4,8,1] => ? = 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [1,4,3,7,2,6,5,8] => [5,3,2,7,6,4,8,1] => ? = 1
[(1,5),(2,7),(3,4),(6,8)]
 => [4,5,7,3,1,8,2,6] => [1,4,3,7,2,5,6,8] => [5,3,2,6,7,4,8,1] => ? = 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,5,7,1,3,8,2,6] => [1,4,2,5,3,7,6,8] => [3,5,2,4,7,6,8,1] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,5,1,7,4,8,2,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,5,7,4,8,3,6] => [1,2,3,5,4,7,6,8] => [2,3,5,4,7,6,8,1] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [1,2,3,5,4,7,6,8] => [2,3,5,4,7,6,8,1] => ? = 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,5,1,7,4,8,6,2] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,5,7,1,3,8,6,2] => [1,4,2,5,3,7,6,8] => [3,5,2,4,7,6,8,1] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [4,5,7,3,1,8,6,2] => [1,4,3,7,6,8,2,5] => [7,3,2,8,5,4,6,1] => ? = 1
[(1,6),(2,8),(3,4),(5,7)]
 => [4,6,7,3,8,1,5,2] => [1,4,3,7,5,8,2,6] => [7,3,2,5,8,4,6,1] => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [1,4,3,7,2,6,5,8] => [5,3,2,7,6,4,8,1] => ? = 1
[(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [1,4,3,7,2,6,5,8] => [5,3,2,7,6,4,8,1] => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [5,3,7,2,6,4,8,1] => ? = 2
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [5,3,7,2,6,4,8,1] => ? = 2
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [1,5,3,7,4,8,2,6] => [7,3,5,2,8,4,6,1] => ? = 2
[(1,5),(2,8),(3,6),(4,7)]
 => [5,6,7,8,1,3,4,2] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2
[(1,4),(2,8),(3,6),(5,7)]
 => [4,6,7,1,8,3,5,2] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,6,1,7,8,4,5,2] => [1,3,2,6,4,7,5,8] => [3,2,5,7,4,6,8,1] => ? = 1
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,6,7,8,4,5,3] => [1,2,3,6,4,7,5,8] => [2,3,5,7,4,6,8,1] => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,6,7,8,4,3,5] => [1,2,3,6,4,7,5,8] => [2,3,5,7,4,6,8,1] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,6,1,7,8,4,2,5] => [1,3,2,6,4,7,5,8] => [3,2,5,7,4,6,8,1] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,6,7,1,8,3,2,5] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,6,7,8,1,3,2,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [5,3,7,2,6,4,8,1] => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [5,3,7,2,6,4,8,1] => ? = 2
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [1,5,3,7,4,8,2,6] => [7,3,5,2,8,4,6,1] => ? = 2
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 1
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [3,2,5,7,4,6,8,1] => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [2,3,5,7,4,6,8,1] => ? = 0
[(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [2,3,5,4,7,6,8,1] => ? = 0
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,5),(2,3),(4,7),(6,8)]
 => [3,5,2,7,1,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,5),(2,3),(4,8),(6,7)]
 => [3,5,2,7,1,8,6,4] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [1,4,2,6,5,8,3,7] => [3,7,2,5,4,8,6,1] => 1
[(1,7),(2,4),(3,8),(5,6)]
 => [4,6,7,2,8,5,1,3] => [1,4,2,6,5,8,3,7] => [3,7,2,5,4,8,6,1] => 1
[(1,3),(2,5),(4,8),(6,7)]
 => [3,5,1,7,2,8,6,4] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1
[(1,4),(2,7),(3,8),(5,6)]
 => [4,6,7,1,8,5,2,3] => [1,4,2,6,5,8,3,7] => [3,7,2,5,4,8,6,1] => 1
[(1,4),(2,8),(3,7),(5,6)]
 => [4,6,7,1,8,5,3,2] => [1,4,2,6,5,8,3,7] => [3,7,2,5,4,8,6,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => 0
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => [1,3,2,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,1] => 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [1,3,2,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,1] => 1

Description

The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern

$213$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.

Matching statistic: St000932

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%

Values

[(1,2)]
 => [2,1] => [2,1] => [1,1,0,0]
 => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[(1,4),(2,3)]
 => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [3,6,2,1,5,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [3,6,2,5,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [3,6,2,5,4,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [4,6,5,2,3,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [4,6,5,2,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,5),(2,3),(4,6),(7,8)]
 => [3,5,2,6,1,4,8,7] => [3,8,2,7,1,6,5,4] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,7),(2,3),(4,5),(6,8)]
 => [3,5,2,7,4,8,1,6] => [3,8,2,7,6,5,1,4] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [3,8,2,7,6,5,4,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,8),(2,4),(3,5),(6,7)]
 => [4,5,7,2,3,8,6,1] => [4,8,7,2,6,5,3,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [4,8,7,2,6,5,1,3] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,6),(2,4),(3,5),(7,8)]
 => [4,5,6,2,3,1,8,7] => [4,8,7,2,6,1,5,3] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
 => [4,5,6,2,1,3,8,7] => [4,8,7,2,1,6,5,3] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,5,1,6,4,2,8,7] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,5,6,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [4,5,6,3,1,2,8,7] => [4,8,7,3,1,6,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [4,8,7,3,2,6,1,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [4,8,7,3,2,6,5,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [4,8,7,3,6,2,5,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [4,8,7,3,6,2,1,5] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [4,8,7,3,6,1,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,5),(2,7),(3,4),(6,8)]
 => [4,5,7,3,1,8,2,6] => [4,8,7,3,1,6,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,5,7,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,5,1,7,4,8,2,6] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,5,7,4,8,3,6] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,5,1,7,4,8,6,2] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,5,7,1,3,8,6,2] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [4,5,7,3,1,8,6,2] => [4,8,7,3,1,6,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,6),(2,8),(3,4),(5,7)]
 => [4,6,7,3,8,1,5,2] => [4,8,7,3,6,1,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [4,8,7,3,6,5,1,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [5,8,7,6,3,4,2,1] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [5,8,7,6,3,1,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,5),(2,8),(3,6),(4,7)]
 => [5,6,7,8,1,3,4,2] => [5,8,7,6,1,4,3,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,4),(2,8),(3,6),(5,7)]
 => [4,6,7,1,8,3,5,2] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,6,1,7,8,4,5,2] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,6,7,8,4,5,3] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,6,7,8,4,3,5] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,6,1,7,8,4,2,5] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,6,7,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,6,7,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [5,8,7,6,3,1,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [5,8,7,6,3,2,1,4] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [5,8,7,6,3,2,4,1] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [5,8,7,6,2,4,3,1] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [5,8,7,6,2,4,1,3] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [5,8,7,6,2,1,4,3] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,4),(2,5),(3,7),(6,8)]
 => [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[(1,2),(3,4),(5,7),(6,8)]
 => [2,1,4,3,7,8,5,6] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,5),(4,8),(6,7)]
 => [2,1,5,7,3,8,6,4] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,6),(4,8),(5,7)]
 => [2,1,6,7,8,3,5,4] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[(1,2),(3,7),(4,8),(5,6)]
 => [2,1,6,7,8,5,3,4] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 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] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Matching statistic: St000054

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%

Values

[(1,2)]
 => [2,1] => [1,2] => [2,1] => 2 = 0 + 2
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 0 + 2
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 3 = 1 + 2
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 3 = 1 + 2
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 2 = 0 + 2
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 3 = 1 + 2
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [3,4,2,5,6,1] => 3 = 1 + 2
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 3 = 1 + 2
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [3,4,5,6,2,1] => 3 = 1 + 2
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 3 = 1 + 2
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [3,5,4,2,6,1] => 3 = 1 + 2
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 3 = 1 + 2
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 3 = 1 + 2
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 2 = 0 + 2
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => 2 = 0 + 2
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [3,2,5,6,4,1] => 3 = 1 + 2
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [3,5,2,6,4,1] => 3 = 1 + 2
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 3 = 1 + 2
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 3 = 1 + 2
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 2 = 0 + 2
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 3 = 1 + 2
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [3,4,2,5,6,7,8,1] => 3 = 1 + 2
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [3,4,5,2,6,7,8,1] => 3 = 1 + 2
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [3,4,5,6,2,7,8,1] => ? = 1 + 2
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [3,4,5,6,7,2,8,1] => ? = 1 + 2
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [3,4,5,6,7,8,2,1] => ? = 1 + 2
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => ? = 1 + 2
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [3,5,4,6,7,2,8,1] => ? = 1 + 2
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [3,5,4,6,2,7,8,1] => ? = 1 + 2
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [3,5,4,2,6,7,8,1] => ? = 1 + 2
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1 + 2
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 1 + 2
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [2,3,5,4,6,7,8,1] => ? = 0 + 2
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [2,3,5,6,4,7,8,1] => ? = 0 + 2
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [3,2,5,6,4,7,8,1] => ? = 1 + 2
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [3,5,2,6,4,7,8,1] => ? = 1 + 2
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [3,5,6,2,4,7,8,1] => ? = 1 + 2
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [3,5,6,4,2,7,8,1] => ? = 1 + 2
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [3,5,6,4,7,2,8,1] => ? = 1 + 2
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => ? = 1 + 2
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => ? = 1 + 2
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [3,5,6,7,4,2,8,1] => ? = 1 + 2
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [3,5,6,7,2,4,8,1] => ? = 1 + 2
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [3,5,6,2,7,4,8,1] => ? = 1 + 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [3,5,2,6,7,4,8,1] => ? = 1 + 2
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [3,2,5,6,7,4,8,1] => ? = 1 + 2
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [2,3,5,6,7,4,8,1] => ? = 0 + 2
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [2,3,5,6,7,8,4,1] => ? = 0 + 2
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [3,2,5,6,7,8,4,1] => ? = 1 + 2
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [3,5,2,6,7,8,4,1] => ? = 1 + 2
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [3,5,6,2,7,8,4,1] => ? = 1 + 2
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [3,5,6,7,2,8,4,1] => ? = 1 + 2
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [3,5,6,7,8,2,4,1] => ? = 1 + 2
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [3,5,6,7,8,4,2,1] => ? = 1 + 2
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [3,5,7,6,8,4,2,1] => ? = 2 + 2
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [3,5,7,6,8,2,4,1] => ? = 2 + 2
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [3,5,7,6,2,8,4,1] => ? = 2 + 2
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [3,5,7,2,6,8,4,1] => ? = 2 + 2
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [3,5,2,7,6,8,4,1] => ? = 1 + 2
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [3,2,5,7,6,8,4,1] => ? = 1 + 2
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [2,3,5,7,6,8,4,1] => ? = 0 + 2
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [2,3,5,7,6,4,8,1] => ? = 0 + 2
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [3,2,5,7,6,4,8,1] => ? = 1 + 2
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [3,5,2,7,6,4,8,1] => ? = 1 + 2
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [3,5,7,2,6,4,8,1] => ? = 2 + 2
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => ? = 2 + 2
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => ? = 2 + 2
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [3,5,7,6,4,8,2,1] => ? = 2 + 2
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [3,5,7,4,6,8,2,1] => ? = 2 + 2
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => ? = 2 + 2
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [3,5,7,4,2,6,8,1] => ? = 2 + 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 2 + 2
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 1 + 2
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 3 = 1 + 2
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => 2 = 0 + 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => [1,3,2,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,1] => 3 = 1 + 2

Description

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation

$\pi$ of

$n$ , together with its rotations, obtained by conjugating with the long cycle

$(1,\dots,n)$ . Drawing the labels

$1$ to

$n$ in this order on a circle, and the arcs

$(i, \pi(i))$ as straight lines, the rotation of

$\pi$ is obtained by replacing each number

$i$ by

$(i\bmod n) +1$ . Then,

$\pi(1)-1$ is the number of rotations of

$\pi$ where the arc

$(1, \pi(1))$ is a deficiency. In particular, if

$O(\pi)$ is the orbit of rotations of

$\pi$ , then the number of deficiencies of

$\pi$ equals

$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$

Matching statistic: St000022
(load all 3 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%

Values

[(1,2)]
 => [2,1] => [2,1] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 1
[(1,4),(2,3)]
 => [3,4,2,1] => [3,4,2,1] => [2,4,3,1] => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,6,5,4,3] => [2,1,4,5,6,3] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,6,1,5,4,2] => [4,5,3,6,1,2] => 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [3,6,2,1,5,4] => [2,5,3,6,1,4] => 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [3,6,2,5,1,4] => [5,6,3,2,1,4] => 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [3,6,2,5,4,1] => [4,5,3,6,2,1] => 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [4,6,5,2,3,1] => [3,5,2,4,6,1] => 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [4,6,5,2,1,3] => [2,5,1,4,6,3] => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,6,5,1,3,2] => [3,5,1,4,6,2] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,6,1,5,4,2] => [4,5,3,6,1,2] => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [2,1,4,5,6,3] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => [2,1,4,5,6,3] => 0
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,6,1,5,4,2] => [4,5,3,6,1,2] => 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,6,5,1,3,2] => [3,5,1,4,6,2] => 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [4,6,5,3,1,2] => [3,1,5,4,6,2] => 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [4,6,5,3,2,1] => [2,3,5,4,6,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => [2,5,3,6,7,8,1,4] => ? = 1
[(1,5),(2,3),(4,6),(7,8)]
 => [3,5,2,6,1,4,8,7] => [3,8,2,7,1,6,5,4] => [5,6,3,7,8,2,1,4] => ? = 1
[(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => [5,6,3,7,2,8,1,4] => ? = 1
[(1,7),(2,3),(4,5),(6,8)]
 => [3,5,2,7,4,8,1,6] => [3,8,2,7,6,5,1,4] => [5,6,3,2,7,8,1,4] => ? = 1
[(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [3,8,2,7,6,5,4,1] => [4,5,3,6,7,8,2,1] => ? = 1
[(1,8),(2,4),(3,5),(6,7)]
 => [4,5,7,2,3,8,6,1] => [4,8,7,2,6,5,3,1] => [3,5,6,4,7,2,8,1] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [4,8,7,2,6,5,1,3] => [5,6,2,4,7,1,8,3] => ? = 1
[(1,6),(2,4),(3,5),(7,8)]
 => [4,5,6,2,3,1,8,7] => [4,8,7,2,6,1,5,3] => [5,6,7,4,2,1,8,3] => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
 => [4,5,6,2,1,3,8,7] => [4,8,7,2,1,6,5,3] => [2,5,6,4,7,1,8,3] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,5,1,6,4,2,8,7] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,5,6,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [4,5,6,3,1,2,8,7] => [4,8,7,3,1,6,5,2] => [5,3,6,4,7,1,8,2] => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => [2,3,6,4,7,1,8,5] => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [4,8,7,3,2,6,1,5] => [6,3,7,4,2,1,8,5] => ? = 1
[(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [4,8,7,3,2,6,5,1] => [5,3,6,4,7,2,8,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [4,8,7,3,6,2,5,1] => [5,6,7,4,3,2,8,1] => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [4,8,7,3,6,2,1,5] => [2,6,7,4,3,1,8,5] => ? = 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [4,8,7,3,6,1,5,2] => [5,6,7,4,3,1,8,2] => ? = 1
[(1,5),(2,7),(3,4),(6,8)]
 => [4,5,7,3,1,8,2,6] => [4,8,7,3,1,6,5,2] => [5,3,6,4,7,1,8,2] => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,5,7,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,5,1,7,4,8,2,6] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,5,7,4,8,3,6] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,5,1,7,4,8,6,2] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,5,7,1,3,8,6,2] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [4,5,7,3,1,8,6,2] => [4,8,7,3,1,6,5,2] => [5,3,6,4,7,1,8,2] => 1
[(1,6),(2,8),(3,4),(5,7)]
 => [4,6,7,3,8,1,5,2] => [4,8,7,3,6,1,5,2] => [5,6,7,4,3,1,8,2] => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [4,8,7,3,6,5,1,2] => [5,1,6,4,7,3,8,2] => 1
[(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => [2,5,6,4,7,3,8,1] => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [5,8,7,6,3,4,2,1] => [2,4,6,3,5,7,8,1] => ? = 2
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => [4,1,6,3,5,7,8,2] => ? = 2
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [5,8,7,6,3,1,4,2] => [4,3,6,1,5,7,8,2] => ? = 2
[(1,5),(2,8),(3,6),(4,7)]
 => [5,6,7,8,1,3,4,2] => [5,8,7,6,1,4,3,2] => [3,4,6,1,5,7,8,2] => ? = 2
[(1,4),(2,8),(3,6),(5,7)]
 => [4,6,7,1,8,3,5,2] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,6,1,7,8,4,5,2] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,6,7,8,4,5,3] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,6,7,8,4,3,5] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,6,1,7,8,4,2,5] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,6,7,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,6,7,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => [3,4,6,1,5,7,8,2] => ? = 2
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [5,8,7,6,3,1,4,2] => [4,3,6,1,5,7,8,2] => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [5,8,7,6,3,2,1,4] => [2,3,6,1,5,7,8,4] => ? = 2
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [5,8,7,6,3,2,4,1] => [4,3,6,2,5,7,8,1] => ? = 2
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [5,8,7,6,2,4,3,1] => [3,4,6,2,5,7,8,1] => ? = 2
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [5,8,7,6,2,4,1,3] => [4,6,2,1,5,7,8,3] => ? = 2
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [5,8,7,6,2,1,4,3] => [2,4,6,1,5,7,8,3] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [3,4,6,1,5,7,8,2] => ? = 2
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => [3,5,6,4,7,1,8,2] => ? = 1
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [4,5,3,6,7,8,1,2] => ? = 1

Description

The number of fixed points of a permutation.