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Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St000920
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St000396
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 75%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> [.,.]
=> [.,.]
=> 1
[1,2] => [1,0,1,0]
=> [.,[.,.]]
=> [[.,.],.]
=> 1
[2,1] => [1,1,0,0]
=> [[.,.],.]
=> [.,[.,.]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> 2
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[[.,.],[[.,.],.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 3
[6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? = 2
[8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> ? = 3
[6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],[[.,.],[.,.]]]
=> ? = 2
[7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> ? = 3
[6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],[[.,.],[.,.]]]
=> ? = 2
[5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,6,8,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[[.,.],[[.,.],.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 3
[6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? = 2
[8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,5,6,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,6,4,5,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,5,4,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 3
[6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ? = 2
[7,5,6,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 3
[6,5,7,4,8,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[[.,.],[.,[.,.]]]]
=> ? = 2
[5,6,7,4,8,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 3
[6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ? = 2
[7,5,4,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 3
[7,4,5,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 3
[6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[5,6,4,7,8,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[[.,.],[[.,.],.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 3
[8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
[7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,[.,.]]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[[.,.],.],[.,.]]]
=> ? = 3
[6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],[[.,.],[.,.]]]
=> ? = 2
[6,5,7,8,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],[[.,.],[.,.]]]
=> ? = 2
[5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St001741
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1,0]
=> [1] => 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,2] => 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,2] => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,2,6,5,7,3] => [1,4,6,7,3,2,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,2,6,7,5,3] => [1,4,6,5,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
[4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,2,7,3,6,5] => [1,4,7,5,3,2,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,2,7,5,3,6] => [1,4,7,6,3,2,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,2,7,5,6,3] => [1,4,7,3,2,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,2,7,6,3,5] => [1,4,7,5,6,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,2,7,6,5,3] => [1,4,7,3,2,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,3,6,2,7,5] => [1,4,6,7,5,2,3] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,3,6,5,7,2] => [1,4,6,7,2,3,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,3,6,7,5,2] => [1,4,6,5,7,2,3] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
[4,1,3,7,2,5,6] => [1,4,7,6,5,2,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,3,7,2,6,5] => [1,4,7,5,2,3,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,3,7,5,2,6] => [1,4,7,6,2,3,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,3,7,5,6,2] => [1,4,7,2,3,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,3,7,6,2,5] => [1,4,7,5,6,2,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,3,7,6,5,2] => [1,4,7,2,3,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,5,6,2,7,3] => [1,4,6,7,3,5,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,5,6,3,7,2] => [1,4,6,7,2,3,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,1,5,6,7,3,2] => [1,4,6,3,5,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => ? = 2
[4,1,5,7,2,3,6] => [1,4,7,6,3,5,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,5,7,2,6,3] => [1,4,7,3,5,2,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,5,7,3,2,6] => [1,4,7,6,2,3,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,5,7,3,6,2] => [1,4,7,2,3,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,5,7,6,2,3] => [1,4,7,3,5,6,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,5,7,6,3,2] => [1,4,7,2,3,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,6,7,2,3,5] => [1,4,7,5,2,3,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,6,7,2,5,3] => [1,4,7,3,6,5,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,6,7,3,2,5] => [1,4,7,5,3,6,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,6,7,3,5,2] => [1,4,7,2,3,6,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,6,7,5,2,3] => [1,4,7,3,6,2,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,6,7,5,3,2] => [1,4,7,2,3,6,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,1,7,6,2,3,5] => [1,4,6,3,7,5,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
[4,1,7,6,3,2,5] => [1,4,6,2,3,7,5] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => ? = 2
[4,1,7,6,3,5,2] => [1,4,6,5,3,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => ? = 2
[4,1,7,6,5,2,3] => [1,4,6,2,3,7,5] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => ? = 2
[4,1,7,6,5,3,2] => [1,4,6,3,7,2,5] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
[4,2,1,6,3,7,5] => [1,4,6,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,1,6,5,7,3] => [1,4,6,7,3,2,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,1,6,7,5,3] => [1,4,6,5,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
[4,2,1,7,3,5,6] => [1,4,7,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,2,1,7,3,6,5] => [1,4,7,5,3,2,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,2,1,7,5,3,6] => [1,4,7,6,3,2,5] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,2,1,7,5,6,3] => [1,4,7,3,2,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,2,1,7,6,3,5] => [1,4,7,5,6,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,2,1,7,6,5,3] => [1,4,7,3,2,5,6] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[4,2,3,6,1,7,5] => [1,4,6,7,5,2,3] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,3,6,5,7,1] => [1,4,6,7,2,3,5] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[4,2,3,6,7,5,1] => [1,4,6,5,7,2,3] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St000485
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ? = 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[3,1,4,5,2,6,7] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[3,1,4,5,2,7,6] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[3,1,4,5,6,2,7] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 2
[3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 2
[3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,7,3,4,6,5,2] => ? = 2
[3,1,4,5,7,6,2] => [1,3,4,5,7,2,6] => [1,6,3,4,7,2,5] => ? = 2
[3,1,4,6,2,5,7] => [1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => ? = 2
[3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => ? = 2
[3,1,4,6,5,2,7] => [1,3,4,6,2,5,7] => [1,5,3,6,2,4,7] => ? = 2
[3,1,4,6,5,7,2] => [1,3,4,6,7,2,5] => [1,6,3,7,5,2,4] => ? = 2
[3,1,4,6,7,2,5] => [1,3,4,6,2,5,7] => [1,5,3,6,2,4,7] => ? = 2
[3,1,4,6,7,5,2] => [1,3,4,6,5,7,2] => [1,7,3,5,4,6,2] => ? = 2
[3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => [1,7,3,6,5,4,2] => ? = 2
[3,1,4,7,2,6,5] => [1,3,4,7,5,2,6] => [1,6,3,7,5,2,4] => ? = 2
[3,1,4,7,5,2,6] => [1,3,4,7,6,2,5] => [1,6,3,7,5,2,4] => ? = 2
[3,1,4,7,5,6,2] => [1,3,4,7,2,5,6] => [1,5,3,7,2,6,4] => ? = 2
[3,1,4,7,6,2,5] => [1,3,4,7,5,6,2] => [1,7,3,6,5,4,2] => ? = 2
[3,1,4,7,6,5,2] => [1,3,4,7,2,5,6] => [1,5,3,7,2,6,4] => ? = 2
[3,1,5,2,4,6,7] => [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[3,1,5,2,4,7,6] => [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[3,1,5,2,6,4,7] => [1,3,5,6,4,2,7] => [1,6,5,4,3,2,7] => ? = 2
[3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => ? = 2
[3,1,5,2,7,4,6] => [1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => ? = 2
[3,1,5,2,7,6,4] => [1,3,5,7,4,2,6] => [1,6,5,7,3,2,4] => ? = 2
[3,1,5,4,6,2,7] => [1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[3,1,5,4,6,7,2] => [1,3,5,6,7,2,4] => [1,6,7,4,5,2,3] => ? = 2
[3,1,5,4,7,2,6] => [1,3,5,7,6,2,4] => [1,6,7,5,4,2,3] => ? = 2
[3,1,5,4,7,6,2] => [1,3,5,7,2,4,6] => [1,5,6,7,2,3,4] => ? = 2
[3,1,5,6,4,2,7] => [1,3,5,4,6,2,7] => [1,6,4,3,5,2,7] => ? = 2
[3,1,5,6,4,7,2] => [1,3,5,4,6,7,2] => [1,7,4,3,5,6,2] => ? = 2
[3,1,5,6,7,2,4] => [1,3,5,7,4,6,2] => [1,7,5,6,3,4,2] => ? = 2
[3,1,5,6,7,4,2] => [1,3,5,7,2,4,6] => [1,5,6,7,2,3,4] => ? = 2
[3,1,5,7,4,2,6] => [1,3,5,4,7,6,2] => [1,7,4,3,6,5,2] => ? = 2
[3,1,5,7,4,6,2] => [1,3,5,4,7,2,6] => [1,6,4,3,7,2,5] => ? = 2
[3,1,5,7,6,2,4] => [1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[3,1,5,7,6,4,2] => [1,3,5,6,4,7,2] => [1,7,5,4,3,6,2] => ? = 2
[3,1,6,2,4,5,7] => [1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 2
[3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => ? = 2
[3,1,6,2,5,4,7] => [1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[3,1,6,2,5,7,4] => [1,3,6,7,4,2,5] => [1,6,7,5,4,2,3] => ? = 2
[3,1,6,2,7,4,5] => [1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[3,1,6,2,7,5,4] => [1,3,6,5,7,4,2] => [1,7,6,4,5,3,2] => ? = 2
[3,1,6,4,2,5,7] => [1,3,6,5,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[3,1,6,4,2,7,5] => [1,3,6,7,5,2,4] => [1,6,7,5,4,2,3] => ? = 2
[3,1,6,4,5,7,2] => [1,3,6,7,2,4,5] => [1,5,7,6,2,4,3] => ? = 2
[3,1,6,4,7,5,2] => [1,3,6,5,7,2,4] => [1,6,7,4,5,2,3] => ? = 2
[3,1,6,5,2,4,7] => [1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => ? = 2
[3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => [1,7,6,5,4,3,2] => ? = 2
[3,1,6,5,4,7,2] => [1,3,6,7,2,4,5] => [1,5,7,6,2,4,3] => ? = 2
Description
The length of the longest cycle of a permutation.
Matching statistic: St001737
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00069: Permutations —complement⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [1,3,4,2] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [1,3,4,2] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,3,4,2] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [1,2,4,3] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [1,2,4,3] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,3,4,5,2] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [1,3,4,5,2] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,3,4,5,2] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [1,3,4,5,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [2,4,5,1,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [2,4,5,1,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [2,4,5,1,3] => 1 = 2 - 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [2,3,5,6,7,1,4] => ? = 2 - 1
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [2,3,5,6,7,1,4] => ? = 2 - 1
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [3,5,6,7,1,2,4] => ? = 2 - 1
[1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [3,5,6,7,1,4,2] => ? = 2 - 1
[1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [3,5,6,7,1,4,2] => ? = 2 - 1
[1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [3,5,6,7,1,4,2] => ? = 2 - 1
[1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [3,5,6,7,1,4,2] => ? = 2 - 1
[1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [2,3,5,6,7,1,4] => ? = 2 - 1
[1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [2,3,5,6,7,1,4] => ? = 2 - 1
[1,2,6,3,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [2,5,6,7,3,1,4] => ? = 2 - 1
[1,2,6,3,5,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,3,7,4,5] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,3,7,5,4] => [1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [2,5,6,7,4,1,3] => ? = 2 - 1
[1,2,6,4,3,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [2,5,6,7,3,1,4] => ? = 2 - 1
[1,2,6,4,5,3,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,4,7,3,5] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,4,7,5,3] => [1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [2,5,6,7,4,1,3] => ? = 2 - 1
[1,2,6,5,3,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,5,4,3,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,5,7,3,4] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,5,7,4,3] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [2,5,6,7,1,3,4] => ? = 2 - 1
[1,2,6,7,3,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [2,5,6,7,1,4,3] => ? = 2 - 1
[1,2,6,7,3,5,4] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [2,5,6,7,3,1,4] => ? = 2 - 1
[1,2,6,7,4,3,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [2,5,6,7,1,4,3] => ? = 2 - 1
[1,2,6,7,4,5,3] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [2,5,6,7,3,1,4] => ? = 2 - 1
[1,2,6,7,5,3,4] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [2,5,6,7,1,4,3] => ? = 2 - 1
[1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [2,5,6,7,1,4,3] => ? = 2 - 1
[1,2,7,3,4,5,6] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [1,5,6,7,4,3,2] => ? = 2 - 1
[1,2,7,3,4,6,5] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [1,5,6,7,3,2,4] => ? = 2 - 1
[1,2,7,3,5,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [1,5,6,7,3,4,2] => ? = 2 - 1
[1,2,7,3,5,6,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [1,5,6,7,2,3,4] => ? = 2 - 1
[1,2,7,3,6,4,5] => [1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [1,5,6,7,4,2,3] => ? = 2 - 1
[1,2,7,3,6,5,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [1,5,6,7,2,3,4] => ? = 2 - 1
[1,2,7,4,3,5,6] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [1,5,6,7,4,3,2] => ? = 2 - 1
[1,2,7,4,3,6,5] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [1,5,6,7,3,2,4] => ? = 2 - 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St000542
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,3,1] => 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [3,4,1,2] => 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [3,4,1,2] => 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [2,4,3,1] => 2
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 2
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [2,4,3,1] => 2
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 2
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 2
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => [2,4,5,1,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,1,2,4] => [2,4,5,1,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [2,3,5,6,1,4,7] => ? = 2
[1,2,5,3,6,7,4] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ? = 2
[1,2,5,3,7,4,6] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [2,3,5,6,1,7,4] => ? = 2
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [2,3,5,6,1,4,7] => ? = 2
[1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ? = 2
[1,2,5,4,7,3,6] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [2,3,5,6,1,7,4] => ? = 2
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2
[1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000541
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [3,4,1,2] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [3,4,1,2] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [2,4,3,1] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [2,4,3,1] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => [2,4,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => [2,4,5,1,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,1,2,4] => [2,4,5,1,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2 - 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2 - 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2 - 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ? = 2 - 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 2 - 1
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [2,3,5,6,1,4,7] => ? = 2 - 1
[1,2,5,3,6,7,4] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ? = 2 - 1
[1,2,5,3,7,4,6] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [2,3,5,6,1,7,4] => ? = 2 - 1
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2 - 1
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [2,3,5,6,1,4,7] => ? = 2 - 1
[1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ? = 2 - 1
[1,2,5,4,7,3,6] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [2,3,5,6,1,7,4] => ? = 2 - 1
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2 - 1
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 2 - 1
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ? = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000062
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 2
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 2
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 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
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 2
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 2
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,3,2] => ? = 2
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,3,2] => ? = 2
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 2
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 2
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000258
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000258: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000258: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => ([(5,6)],7)
=> ?
=> ? = 2
Description
The burning number of a graph.
This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.
Matching statistic: St000299
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000299: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000299: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,4,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,5,4,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => ([],7)
=> ?
=> ? = 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,3,5,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,5,3,6] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ?
=> ? = 2
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => ([(5,6)],7)
=> ?
=> ? = 2
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => ([(5,6)],7)
=> ?
=> ? = 2
Description
The number of nonisomorphic vertex-induced subtrees.
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001734The lettericity of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000397The Strahler number of a rooted tree. St000535The rank-width of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank 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 $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001555The order of a signed permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001624The breadth of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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