Your data matches 68 different statistics following compositions of up to 3 maps.
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Mp00229: Dyck paths Delest-ViennotDyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Mp00229: Dyck paths Delest-ViennotDyck paths
Mp00140: Dyck paths logarithmic height to pruning numberBinary trees
St000396: Binary trees ⟶ ℤResult quality: 56% values known / values provided: 56%distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [.,.]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[[[[.,[[[.,.],.],.]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,[[.,[.,[.,.]]],.]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[[[[.,.],[[[.,.],.],.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[[[.,[[[.,.],.],[.,.]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,[[.,.],[.,[.,.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[[[.,[[.,.],.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,[.,[[.,.],[.,.]]]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[.,[.,[.,.]]],.],.]],.],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[[.,[[[.,[.,.]],[.,.]],.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[.,.],[[[[.,.],.],.],.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[[[.,.],[[[.,.],[.,.]],.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,[.,[.,.]]]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[[[.,.],[[[.,.],.],[.,.]]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,[.,[[[[.,.],.],.],.]]],.],.]
=> ? = 1
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.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
St000485: Permutations ⟶ ℤResult quality: 27% values known / values provided: 27%distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,5,2,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,4,5,6,2,7] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,5,2,7,6] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,7,3,4,6,5,2] => ? = 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,4,5,2,6,7] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,4,6,7,2,5] => [1,3,4,6,2,5,7] => [1,5,3,6,2,4,7] => ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5,7] => [1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => ? = 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => [1,7,3,6,5,4,2] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => [1,3,5,6,7,2,4] => [1,6,7,4,5,2,3] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => [1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => [1,3,5,7,2,4,6] => [1,5,6,7,2,3,4] => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => [1,3,5,7,6,2,4] => [1,6,7,5,4,2,3] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,4,6,1,2,7,5] => [1,3,6,7,5,2,4] => [1,6,7,5,4,2,3] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,6,7,1,2,5] => [1,3,6,2,4,7,5] => [1,4,7,2,5,6,3] => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5,7] => [1,3,6,5,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,7,1,2,5,6] => [1,3,7,6,5,2,4] => [1,6,7,5,4,2,3] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,7] => [1,3,5,6,4,2,7] => [1,6,5,4,3,2,7] => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => [1,3,5,7,4,6,2] => [1,7,5,6,3,4,2] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,7,6] => [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,6] => [1,3,5,7,6,4,2] => [1,7,6,5,4,3,2] => ? = 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,5,2,4,6,7] => [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => [1,3,6,7,4,2,5] => [1,6,7,5,4,2,3] => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => [1,3,6,2,5,7,4] => [1,4,7,2,5,6,3] => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => [1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => [1,3,7,6,4,2,5] => [1,6,7,5,4,2,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => [1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,6,7,2,4,5] => [1,3,6,4,7,5,2] => [1,7,6,4,5,3,2] => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,6,2,4,5,7] => [1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => [1,3,7,5,2,6,4] => [1,5,7,6,2,4,3] => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => ? = 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => [1,4,5,6,7,3,2] => [1,7,6,4,5,3,2] => ? = 2
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,6,3,7] => [1,4,5,6,3,2,7] => [1,6,5,4,3,2,7] => ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,3,7,6] => [1,4,5,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [4,1,2,5,7,3,6] => [1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => ? = 2
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,3,6,7] => [1,4,5,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => [1,4,6,7,3,5,2] => [1,7,6,5,4,3,2] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => [1,4,6,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => [1,4,7,6,3,5,2] => [1,7,6,5,4,3,2] => ? = 2
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,5,1,2,6,7,3] => [1,4,2,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 2
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [4,5,1,2,6,3,7] => [1,4,2,5,6,3,7] => [1,6,3,4,5,2,7] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => [1,4,6,7,3,2,5] => [1,6,7,5,4,2,3] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => [1,4,6,2,5,7,3] => [1,7,5,4,3,6,2] => ? = 2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => [1,4,6,3,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => [1,4,2,5,7,3,6] => [1,6,3,4,7,2,5] => ? = 2
Description
The length of the longest cycle of a permutation.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
St000058: Permutations ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,5,2,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,5,6,3,7] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,4,1,5,7,3,6] => [1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => [1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [1,2,5,6,4,3,7] => [1,2,6,5,4,3,7] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => [1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => [1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => [1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [1,2,5,3,7,6,4] => [1,2,7,4,6,5,3] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,3,7,4,5] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,6,1,3,7,4,5] => [1,2,6,4,3,5,7] => [1,2,6,5,4,3,7] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => [1,2,7,5,4,6,3] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,6,2,7,4,5] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ? = 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,4,5,6,2,7] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,5,2,7,6] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => [1,7,3,4,6,5,2] => ? = 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,4,5,2,6,7] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,6,7,5] => [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,7] => [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,4,6,7,2,5] => [1,3,4,6,2,5,7] => [1,5,3,6,2,4,7] => ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5,7] => [1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,2,5,7,6] => [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,4,2,7,5,6] => [1,3,4,2,5,7,6] => [1,4,3,2,5,7,6] => ? = 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4,2,5,6,7] => [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,5,1,6,2,7] => [1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => [1,3,6,2,4,5,7] => [1,4,6,2,5,3,7] => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5,7] => [1,3,6,5,2,4,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4,7] => [1,3,5,6,4,2,7] => [1,6,5,4,3,2,7] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,7,6] => [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,5,2,4,6,7] => [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [3,5,1,2,6,7,4] => [1,3,2,5,6,7,4] => [1,3,2,7,5,6,4] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [3,5,1,2,6,4,7] => [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,5,6,7,1,2,4] => [1,3,6,2,5,4,7] => [1,4,6,2,5,3,7] => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => [1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,7,4,6] => [1,3,2,5,7,6,4] => [1,3,2,7,6,5,4] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => [1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,6,2,4,5,7] => [1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [3,6,1,2,4,7,5] => [1,3,2,6,7,5,4] => [1,3,2,7,6,5,4] => ? = 2
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St001741: Permutations ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [1,2,3] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [1,4,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => [1,6,7,2,3,4,5] => ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => [1,5,6,2,3,4,7] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => [1,5,7,2,3,4,6] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,6,2,3,4,7,5] => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,6,2,3,4,7,5] => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [1,7,2,3,4,6,5] => [1,6,7,2,3,4,5] => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => [1,5,6,7,2,3,4] => ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [1,7,2,3,4,6,5] => [1,6,7,2,3,4,5] => ? = 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [1,7,2,4,3,5,6] => [1,4,7,2,3,5,6] => ? = 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [1,7,2,4,3,5,6] => [1,4,7,2,3,5,6] => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => [1,5,2,3,6,4,7] => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [1,7,2,5,3,4,6] => [1,5,2,3,7,4,6] => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => [1,6,2,3,7,4,5] => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [1,7,2,6,3,4,5] => [1,6,2,3,7,4,5] => ? = 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => [1,5,2,3,6,4,7] => ? = 2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,1,7] => [1,7,2,5,3,4,6] => [1,5,2,3,7,4,6] => ? = 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [1,7,2,6,3,4,5] => [1,6,2,3,7,4,5] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => [1,7,2,6,3,5,4] => [1,6,2,3,5,7,4] => ? = 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [1,7,2,6,3,4,5] => [1,6,2,3,7,4,5] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => [1,5,6,2,3,4,7] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,1,7] => [1,7,2,3,5,4,6] => [1,5,7,2,3,4,6] => ? = 2
Description
The largest integer such that all patterns of this size are contained in the permutation.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00065: Permutations permutation posetPosets
St000298: Posets ⟶ ℤResult quality: 21% values known / values provided: 21%distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,6,7,3,5] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,5,6,3,7] => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,7,3] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,5,1,6,3,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [1,2,4,7,6,3,5] => ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,7,3,5] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,6,1,3,7,5] => [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,6,1,7,3,5] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => [1,2,4,7,5,3,6] => ([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,6,7,3,4] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,1,5,7,3,4,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => [1,2,5,7,4,6,3] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => [1,2,5,3,6,7,4] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => [1,2,5,7,4,3,6] => ([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => [1,2,5,3,4,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [1,2,5,3,7,6,4] => ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00069: Permutations complementPermutations
St000862: Permutations ⟶ ℤResult quality: 21% values known / values provided: 21%distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [5,4,2,3,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [5,4,1,3,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,5,1] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,6,1] => [1,2,6,3,5,4,7] => [7,6,2,5,3,4,1] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 2
Description
The number of parts of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of parts of the shifted shape.
Matching statistic: St001876
Mp00229: Dyck paths Delest-ViennotDyck paths
Mp00232: Dyck paths parallelogram posetPosets
Mp00195: Posets order idealsLattices
St001876: Lattices ⟶ ℤResult quality: 15% values known / values provided: 15%distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,8),(2,10),(3,10),(4,9),(5,9),(6,7),(7,2),(7,3),(8,4),(8,5),(9,6),(10,1)],11)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,8),(1,10),(2,11),(4,9),(5,3),(6,4),(6,11),(7,5),(8,2),(8,6),(9,10),(10,7),(11,1),(11,9)],12)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,8),(2,9),(3,9),(4,10),(5,10),(6,1),(7,4),(7,5),(8,2),(8,3),(9,7),(10,6)],11)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,8),(2,9),(3,9),(4,10),(5,10),(6,1),(7,4),(7,5),(8,2),(8,3),(9,7),(10,6)],11)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 2 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St001060: Graphs ⟶ ℤResult quality: 15% values known / values provided: 15%distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Mp00330: Dyck paths rotate triangulation clockwiseDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00160: Permutations graph of inversionsGraphs
St000264: Graphs ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 25%
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]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 2 + 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]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 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]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [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
[1,0,1,1,0,1,0,1,0,0]
=> [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
[1,0,1,1,0,1,1,0,0,0]
=> [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
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [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
[1,0,1,1,1,0,1,0,0,0]
=> [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
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [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
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [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]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 2 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [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
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [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
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [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
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [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
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [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
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [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
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [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
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [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
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [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
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [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
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [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
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [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
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [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
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [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
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [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
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [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
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [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
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [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
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [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
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [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
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [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
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [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
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [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
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [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]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [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
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [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
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [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
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [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
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [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
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [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
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [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
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [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
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [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
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [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
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [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
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [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
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001792The arboricity of a graph. St000308The height of the tree associated to a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000397The Strahler number of a rooted tree. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. 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. 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. St001638The book thickness of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000454The largest eigenvalue of a graph if it is integral. St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000486The number of cycles of length at least 3 of a permutation. St000779The tier of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000640The rank of the largest boolean interval in a poset. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000028The number of stack-sorts needed to sort a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000805The number of peaks of the associated bargraph. St001948The number of augmented double ascents of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.