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Your data matches 66 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000701
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
Description
The protection number of a binary tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000920
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 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 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[[.,.],.]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[[.,.],.]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[[.,.],.]]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[[[.,.],.],.]]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[[[.,.],.],.]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[.,[.,.]]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[.,[.,.]]]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[.,[.,.]]]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[.,[.,.]]]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[.,[.,.]]]
=> ? = 2
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: St000353
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,1,4,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,1,4,5,6] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,1,4,5,6] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,2,3,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,2,3,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,1,2,3,6] => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => ? = 2 - 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000779
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,5,1,2,6,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => ? = 2 - 1
Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St001359
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,1,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,4,3,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,2,4,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,4,3,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,2,4,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,2,5,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,2,5,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [2,5,4,3,6,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [2,6,5,4,3,7,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [2,7,6,5,4,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [3,2,4,7,6,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [3,2,5,4,6,7,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [3,2,5,4,7,6,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => ? = 2
Description
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.
In other words, this is $2^k$ where $k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition.
The generating function for the number of equivalence classes, $f(n)$, is
$$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$
Matching statistic: St000486
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,2,1] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,1,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,2,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,2,5,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,2,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,2,5,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,2,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,1,4,5,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [2,5,4,3,6,7,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [2,6,5,4,3,7,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [2,7,6,5,4,3,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [3,2,4,7,6,5,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [3,2,5,4,6,7,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [3,2,5,4,7,6,1] => ? = 2 - 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000264
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> ? = 1 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ? = 1 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 1 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 2 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 1 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 1 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 1 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 1 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001685
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,1,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,4,5,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,3,4,5,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,7,6,5,4,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,6,5,3,4,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,7,6,5,4,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,7,6,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,6,5,3,4,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,7,5,6,4,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,7,5,6,4,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,7,5,6,4,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,7,6,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,7,4,5,6,3,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [1,7,3,4,5,6,2] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,7,5,4,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,6,7,5,3,4,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,7,5,4,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,6,7,5,2,3,4] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [1,6,7,4,5,3,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,6,7,4,5,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,6,7,5,3,4,2] => ? = 2 - 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000646
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,1,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,4,5,3,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,3,4,5,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,4,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,5,1,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,7,6,5,4,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,6,5,3,4,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,7,6,5,4,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,7,6,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,6,5,3,4,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,7,5,6,4,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,7,5,6,4,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,7,5,6,4,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,7,6,4,5,3,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,7,4,5,6,3,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [1,7,3,4,5,6,2] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,6,7,5,4,3,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,7,5,4,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,6,7,5,3,4,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,7,5,4,2,3] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,6,7,5,2,3,4] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [1,6,7,4,5,3,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,6,7,4,5,2,3] => ? = 2 - 1
Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see [[St000441]].
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St000023The number of inner peaks of a permutation. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000397The Strahler number of a rooted tree. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001638The book thickness of a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001335The cardinality of a minimal cycle-isolating set of a graph. St000544The cop number of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001597The Frobenius rank of a skew partition. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001358The largest degree of a regular subgraph of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000906The length of the shortest maximal chain in a poset. St000640The rank of the largest boolean interval in a poset. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001734The lettericity of a graph. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000630The length of the shortest palindromic decomposition of a binary word. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000237The number of small exceedances.
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