Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 2
[2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2
[3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,1,3,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St000396
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00140: Dyck paths logarithmic height to pruning numberBinary trees
St000396: Binary trees ⟶ ℤResult quality: 38% values known / values provided: 38%distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1,0]
=> [.,.]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> 2
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[4,1,3,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> 2
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1
[7,8,6,5,4,3,2,1] => [2,3,4,5,6,8,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 2
[8,6,7,5,4,3,2,1] => [2,3,4,5,7,8,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> ? = 2
[7,6,8,5,4,3,2,1] => [2,3,4,5,8,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> ? = 2
[6,7,8,5,4,3,2,1] => [2,3,4,5,8,6,7,1] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> ? = 2
[7,8,5,6,4,3,2,1] => [2,3,4,6,8,5,7,1] => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[[[.,[[.,.],.]],[.,.]],.],.],.]
=> ? = 2
[8,6,5,7,4,3,2,1] => [2,3,4,7,6,8,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> ? = 2
[8,5,6,7,4,3,2,1] => [2,3,4,7,8,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 2
[7,6,5,8,4,3,2,1] => [2,3,4,8,6,7,5,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[6,7,5,8,4,3,2,1] => [2,3,4,8,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[7,5,6,8,4,3,2,1] => [2,3,4,8,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[6,5,7,8,4,3,2,1] => [2,3,4,8,6,5,7,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[5,6,7,8,4,3,2,1] => [2,3,4,8,5,6,7,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[8,7,6,4,5,3,2,1] => [2,3,5,6,4,7,8,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[.,[.,.]],[[[.,.],.],.]],.],.]
=> ? = 2
[8,6,5,4,7,3,2,1] => [2,3,7,5,6,8,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> ? = 2
[7,6,5,4,8,3,2,1] => [2,3,8,5,6,7,4,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[6,7,5,4,8,3,2,1] => [2,3,8,5,7,6,4,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[7,5,6,4,8,3,2,1] => [2,3,8,6,7,5,4,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[6,5,7,4,8,3,2,1] => [2,3,8,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[7,6,4,5,8,3,2,1] => [2,3,8,6,4,7,5,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[7,5,4,6,8,3,2,1] => [2,3,8,5,7,4,6,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[7,4,5,6,8,3,2,1] => [2,3,8,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[6,5,4,7,8,3,2,1] => [2,3,8,5,6,4,7,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[5,6,4,7,8,3,2,1] => [2,3,8,6,5,4,7,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[6,4,5,7,8,3,2,1] => [2,3,8,6,4,5,7,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[5,4,6,7,8,3,2,1] => [2,3,8,5,4,6,7,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[4,5,6,7,8,3,2,1] => [2,3,8,4,5,6,7,1] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> ? = 2
[8,7,6,5,3,4,2,1] => [2,4,5,3,6,7,8,1] => [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[[.,[.,.]],[[[[.,.],.],.],.]],.]
=> ? = 2
[6,7,5,8,3,4,2,1] => [2,4,8,3,7,6,5,1] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],[[.,.],.]],.]
=> ? = 2
[6,5,7,8,3,4,2,1] => [2,4,8,3,6,5,7,1] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],[[.,.],.]],.]
=> ? = 2
[5,6,7,8,3,4,2,1] => [2,4,8,3,5,6,7,1] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],[[.,.],.]],.]
=> ? = 2
[8,6,7,4,3,5,2,1] => [2,5,4,7,3,8,6,1] => [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [[[[.,[[.,[.,.]],.]],.],[.,.]],.]
=> ? = 2
[8,6,7,3,4,5,2,1] => [2,5,7,3,4,8,6,1] => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
=> ? = 2
[6,7,8,3,4,5,2,1] => [2,5,8,3,4,6,7,1] => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> ? = 2
[7,8,3,4,5,6,2,1] => [2,6,8,3,4,5,7,1] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> ? = 2
[8,6,5,4,3,7,2,1] => [2,7,4,5,6,8,3,1] => [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],[[.,.],.]],.]
=> ? = 2
[8,6,4,3,5,7,2,1] => [2,7,4,6,3,8,5,1] => [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],[[.,.],.]],.]
=> ? = 2
[8,5,4,3,6,7,2,1] => [2,7,4,5,8,3,6,1] => [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> ? = 2
[8,4,3,5,6,7,2,1] => [2,7,4,8,3,5,6,1] => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,[.,.]],.]],.]
=> ? = 2
[8,3,4,5,6,7,2,1] => [2,7,8,3,4,5,6,1] => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> ? = 2
[7,6,5,4,3,8,2,1] => [2,8,4,5,6,7,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[7,5,6,4,3,8,2,1] => [2,8,4,6,7,5,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[6,7,4,5,3,8,2,1] => [2,8,5,7,4,6,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[7,5,4,6,3,8,2,1] => [2,8,6,5,7,4,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[7,4,5,6,3,8,2,1] => [2,8,6,7,4,5,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[6,5,4,7,3,8,2,1] => [2,8,7,5,6,4,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[6,4,5,7,3,8,2,1] => [2,8,7,6,4,5,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[5,4,6,7,3,8,2,1] => [2,8,7,5,4,6,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[4,5,6,7,3,8,2,1] => [2,8,7,4,5,6,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[7,6,5,3,4,8,2,1] => [2,8,5,3,6,7,4,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
Description
The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000485
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
St000485: Permutations ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,3,4,1] => [4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 2
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,4,3,1] => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 2
[3,2,1,4] => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[3,2,4,1] => [4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,1,2] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,3,2] => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[4,2,3,1] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,5,4,2] => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,3,2] => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[2,3,4,5,1,6,7] => [5,2,3,4,1,6,7] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 2
[2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 2
[2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 2
[2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 3
[2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [1,7,6,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 3
[2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 2
[2,3,4,6,1,5,7] => [6,2,3,4,1,5,7] => [1,6,5,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,4,6,1,7,5] => [7,2,3,4,6,1,5] => [1,7,5,6,2,3,4] => [1,7,6,5,4,3,2] => ? = 3
[2,3,4,6,5,1,7] => [5,2,3,4,6,1,7] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,4,6,5,7,1] => [7,2,3,4,6,5,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 3
[2,3,4,6,7,1,5] => [7,2,3,4,1,6,5] => [1,7,5,2,3,4,6] => [1,7,6,4,5,3,2] => ? = 3
[2,3,4,6,7,5,1] => [5,2,3,4,7,6,1] => [1,5,7,2,3,4,6] => [1,6,7,4,5,2,3] => ? = 2
[2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => [1,7,6,5,2,3,4] => [1,7,6,5,4,3,2] => ? = 3
[2,3,4,7,1,6,5] => [6,2,3,4,7,1,5] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 2
[2,3,4,7,5,1,6] => [5,2,3,4,7,1,6] => [1,5,7,6,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,4,7,5,6,1] => [6,2,3,4,7,5,1] => [1,6,5,7,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,4,7,6,1,5] => [6,2,3,4,1,7,5] => [1,6,7,5,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,4,7,6,5,1] => [5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,1,4,6,7] => [5,2,3,1,4,6,7] => [1,5,4,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 2
[2,3,5,1,4,7,6] => [5,2,3,1,4,7,6] => [1,5,4,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 2
[2,3,5,1,6,4,7] => [6,2,3,5,1,4,7] => [1,6,4,5,2,3,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,5,1,6,7,4] => [7,2,3,5,1,6,4] => [1,7,4,5,2,3,6] => [1,7,6,5,4,3,2] => ? = 3
[2,3,5,1,7,4,6] => [7,2,3,5,1,4,6] => [1,7,6,4,5,2,3] => [1,7,6,5,4,3,2] => ? = 3
[2,3,5,1,7,6,4] => [6,2,3,5,1,7,4] => [1,6,7,4,5,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,4,1,6,7] => [4,2,3,5,1,6,7] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 2
[2,3,5,4,1,7,6] => [4,2,3,5,1,7,6] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 2
[2,3,5,4,6,1,7] => [6,2,3,5,4,1,7] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 2
[2,3,5,4,6,7,1] => [7,2,3,5,4,6,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 3
[2,3,5,4,7,1,6] => [7,2,3,5,4,1,6] => [1,7,6,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 3
[2,3,5,4,7,6,1] => [6,2,3,5,4,7,1] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 2
[2,3,5,6,1,4,7] => [6,2,3,1,5,4,7] => [1,6,4,2,3,5,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,5,6,1,7,4] => [7,2,3,6,5,1,4] => [1,7,4,6,2,3,5] => [1,7,6,5,4,3,2] => ? = 3
[2,3,5,6,4,1,7] => [4,2,3,6,5,1,7] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 2
[2,3,5,6,4,7,1] => [7,2,3,6,5,4,1] => [1,7,2,3,4,6,5] => [1,7,3,4,5,6,2] => ? = 3
[2,3,5,6,7,1,4] => [7,2,3,1,5,6,4] => [1,7,4,2,3,5,6] => [1,7,5,4,3,6,2] => ? = 3
[2,3,5,6,7,4,1] => [4,2,3,7,5,6,1] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 2
[2,3,5,7,1,4,6] => [7,2,3,1,5,4,6] => [1,7,6,4,2,3,5] => [1,7,6,5,4,3,2] => ? = 3
[2,3,5,7,1,6,4] => [6,2,3,7,5,1,4] => [1,6,2,3,4,7,5] => [1,7,3,4,5,6,2] => ? = 2
[2,3,5,7,4,1,6] => [4,2,3,7,5,1,6] => [1,4,7,6,2,3,5] => [1,6,7,5,4,2,3] => ? = 2
[2,3,5,7,4,6,1] => [6,2,3,7,5,4,1] => [1,6,4,7,2,3,5] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,7,6,1,4] => [6,2,3,1,5,7,4] => [1,6,7,4,2,3,5] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,7,6,4,1] => [4,2,3,6,5,7,1] => [1,4,6,7,2,3,5] => [1,6,7,5,4,2,3] => ? = 2
[2,3,6,1,4,5,7] => [6,2,3,1,4,5,7] => [1,6,5,4,2,3,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,6,1,4,7,5] => [7,2,3,1,6,4,5] => [1,7,5,6,4,2,3] => [1,7,6,5,4,3,2] => ? = 3
[2,3,6,1,5,4,7] => [5,2,3,6,1,4,7] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 2
[2,3,6,1,5,7,4] => [7,2,3,6,1,5,4] => [1,7,4,6,5,2,3] => [1,7,6,5,4,3,2] => ? = 3
[2,3,6,1,7,4,5] => [7,2,3,6,1,4,5] => [1,7,5,2,3,4,6] => [1,7,6,4,5,3,2] => ? = 3
[2,3,6,1,7,5,4] => [5,2,3,7,6,1,4] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,6,4,1,5,7] => [4,2,3,6,1,5,7] => [1,4,6,5,2,3,7] => [1,6,5,4,3,2,7] => ? = 2
Description
The length of the longest cycle of a permutation.
Mp00064: Permutations reversePermutations
Mp00223: Permutations runsortPermutations
Mp00064: Permutations reversePermutations
St000308: Permutations ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [2,1] => 1
[1,2,3] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[3,1,2] => [2,1,3] => [1,3,2] => [2,3,1] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 2
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 2
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 2
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 2
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 2
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 2
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 2
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Mp00064: Permutations reversePermutations
Mp00223: Permutations runsortPermutations
Mp00071: Permutations descent compositionInteger compositions
St001235: Integer compositions ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => [2] => 1
[2,1] => [1,2] => [1,2] => [2] => 1
[1,2,3] => [3,2,1] => [1,2,3] => [3] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [3] => 1
[2,1,3] => [3,1,2] => [1,2,3] => [3] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1] => 2
[3,1,2] => [2,1,3] => [1,3,2] => [2,1] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [3] => 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4] => 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4] => 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,1] => 2
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,1] => 2
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4] => 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4] => 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [2,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [2,2] => 2
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,2] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [3,1] => 2
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [2,2] => 2
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,2] => 2
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4] => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [2,2] => 2
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,2] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,1] => 2
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [2,2] => 2
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [2,2] => 2
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,1] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,2] => 2
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [3,1] => 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,1] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [3,2] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [3,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [3,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [4,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [3,2] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [3,2] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [3,2] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [3,2] => 2
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [6,1] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [6,1] => ? = 2
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 2
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,2] => ? = 2
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 2
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 2
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [5,2] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,2] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [5,2] => ? = 2
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [4,3] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,2] => ? = 2
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [6,1] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
St001174: Permutations ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 50%
Values
[1] => ? = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 1 = 2 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 1 = 2 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 1 = 2 - 1
[1,4,5,3,2] => 1 = 2 - 1
[1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,5,7,6] => ? = 1 - 1
[1,2,3,4,6,5,7] => ? = 1 - 1
[1,2,3,4,6,7,5] => ? = 2 - 1
[1,2,3,4,7,5,6] => ? = 2 - 1
[1,2,3,4,7,6,5] => ? = 1 - 1
[1,2,3,5,4,6,7] => ? = 1 - 1
[1,2,3,5,4,7,6] => ? = 1 - 1
[1,2,3,5,6,4,7] => ? = 2 - 1
[1,2,3,5,6,7,4] => ? = 2 - 1
[1,2,3,5,7,4,6] => ? = 2 - 1
[1,2,3,5,7,6,4] => ? = 2 - 1
[1,2,3,6,4,5,7] => ? = 2 - 1
[1,2,3,6,4,7,5] => ? = 2 - 1
[1,2,3,6,5,4,7] => ? = 1 - 1
[1,2,3,6,5,7,4] => ? = 2 - 1
[1,2,3,6,7,4,5] => ? = 2 - 1
[1,2,3,6,7,5,4] => ? = 2 - 1
[1,2,3,7,4,5,6] => ? = 2 - 1
[1,2,3,7,4,6,5] => ? = 2 - 1
[1,2,3,7,5,4,6] => ? = 2 - 1
[1,2,3,7,5,6,4] => ? = 2 - 1
[1,2,3,7,6,4,5] => ? = 2 - 1
[1,2,3,7,6,5,4] => ? = 1 - 1
[1,2,4,3,5,6,7] => ? = 1 - 1
[1,2,4,3,5,7,6] => ? = 1 - 1
[1,2,4,3,6,5,7] => ? = 1 - 1
[1,2,4,3,6,7,5] => ? = 2 - 1
[1,2,4,3,7,5,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => ? = 1 - 1
[1,2,4,5,3,6,7] => ? = 2 - 1
[1,2,4,5,3,7,6] => ? = 2 - 1
[1,2,4,5,6,3,7] => ? = 2 - 1
[1,2,4,5,6,7,3] => ? = 2 - 1
[1,2,4,5,7,3,6] => ? = 2 - 1
[1,2,4,5,7,6,3] => ? = 2 - 1
[1,2,4,6,3,5,7] => ? = 2 - 1
[1,2,4,6,3,7,5] => ? = 2 - 1
[1,2,4,6,5,3,7] => ? = 2 - 1
[1,2,4,6,5,7,3] => ? = 2 - 1
[1,2,4,6,7,3,5] => ? = 2 - 1
[1,2,4,6,7,5,3] => ? = 2 - 1
[1,2,4,7,3,5,6] => ? = 2 - 1
[1,2,4,7,3,6,5] => ? = 2 - 1
[1,2,4,7,5,3,6] => ? = 2 - 1
[1,2,4,7,5,6,3] => ? = 2 - 1
[1,2,4,7,6,3,5] => ? = 2 - 1
[1,2,4,7,6,5,3] => ? = 2 - 1
[1,2,5,3,4,6,7] => ? = 2 - 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Mp00064: Permutations reversePermutations
Mp00223: Permutations runsortPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St001859: Permutations ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,4,1,2] => 1 = 2 - 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,1,2,4] => 1 = 2 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 2 - 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 2 - 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 2 - 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 2 - 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 2 - 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 2 - 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 2 - 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1 - 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 2 - 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 2 - 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 2 - 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 2 - 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 2 - 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 2 - 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 2 - 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 - 1
Description
The number of factors of the Stanley symmetric function associated with a permutation. For example, the Stanley symmetric function of $\pi=321645$ equals $20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00154: Graphs coreGraphs
St000264: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1 + 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 1 + 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 1 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,2,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,5,4,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,3,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,5,3,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,4,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,3,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,3,4,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,3,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,5,4,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,5,4,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,5,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,2,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00154: Graphs coreGraphs
St001570: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1 - 2
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 1 - 2
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 1 - 2
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 - 2
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 1 - 2
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,5,2,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,5,4,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,5,3,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,4,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,3,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,3,4,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,4,3,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,1,5,4,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 2
[3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,5,4,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,1,5,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
[4,1,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 2
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00232: Dyck paths parallelogram posetPosets
St000298: Posets ⟶ ℤResult quality: 1% values known / values provided: 1%distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,1,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [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)
=> ? = 2
[1,3,5,2,4] => [1,5,3,2,4] => [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)
=> ? = 2
[1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,4,2,5,3] => [1,4,5,2,3] => [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)
=> ? = 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [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)
=> ? = 2
[1,4,5,2,3] => [1,5,2,4,3] => [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)
=> ? = 2
[1,4,5,3,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [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)
=> ? = 2
[1,5,2,4,3] => [1,5,4,2,3] => [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)
=> ? = 2
[1,5,3,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,5,3,4,2] => [1,4,5,3,2] => [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)
=> ? = 2
[1,5,4,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,5,4,3,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1,5] => [4,2,3,1,5] => [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)
=> ? = 2
[2,3,4,5,1] => [5,2,3,4,1] => [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)
=> ? = 2
[2,3,5,1,4] => [5,2,3,1,4] => [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)
=> ? = 2
[2,3,5,4,1] => [4,2,3,5,1] => [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)
=> ? = 2
[2,4,1,3,5] => [4,2,1,3,5] => [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)
=> ? = 2
[2,4,1,5,3] => [4,2,5,1,3] => [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)
=> ? = 2
[2,4,3,5,1] => [5,2,4,3,1] => [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)
=> ? = 2
[2,4,5,1,3] => [5,2,1,4,3] => [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)
=> ? = 2
[2,5,1,3,4] => [5,2,1,3,4] => [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)
=> ? = 2
[2,5,1,4,3] => [5,2,4,1,3] => [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)
=> ? = 2
[2,5,3,4,1] => [4,2,5,3,1] => [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)
=> ? = 2
[2,5,4,1,3] => [4,2,1,5,3] => [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)
=> ? = 2
[3,1,4,2,5] => [3,4,1,2,5] => [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)
=> ? = 2
[3,1,4,5,2] => [3,5,1,4,2] => [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)
=> ? = 2
[3,1,5,2,4] => [3,5,1,2,4] => [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)
=> ? = 2
[3,1,5,4,2] => [3,4,1,5,2] => [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)
=> ? = 2
[3,2,4,1,5] => [4,3,2,1,5] => [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)
=> ? = 2
[3,2,4,5,1] => [5,3,2,4,1] => [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)
=> ? = 2
[3,2,5,1,4] => [5,3,2,1,4] => [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)
=> ? = 2
[3,2,5,4,1] => [4,3,2,5,1] => [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)
=> ? = 2
[3,4,1,2,5] => [4,1,3,2,5] => [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)
=> ? = 2
[3,4,1,5,2] => [4,5,3,1,2] => [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)
=> ? = 2
[3,4,2,5,1] => [5,4,3,2,1] => [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)
=> ? = 2
[3,4,5,1,2] => [5,1,3,4,2] => [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)
=> ? = 2
[3,4,5,2,1] => [2,5,3,4,1] => [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)
=> ? = 2
[3,5,1,2,4] => [5,1,3,2,4] => [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)
=> ? = 2
[3,5,1,4,2] => [5,4,3,1,2] => [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)
=> ? = 2
[3,5,2,1,4] => [2,5,3,1,4] => [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)
=> ? = 2
[3,5,2,4,1] => [4,5,3,2,1] => [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)
=> ? = 2
[3,5,4,1,2] => [4,1,3,5,2] => [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)
=> ? = 2
[4,1,2,3,5] => [4,1,2,3,5] => [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)
=> ? = 2
[4,1,2,5,3] => [4,1,5,2,3] => [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)
=> ? = 2
[4,1,3,2,5] => [4,3,1,2,5] => [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)
=> ? = 2
[4,1,3,5,2] => [4,5,1,3,2] => [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)
=> ? = 2
[4,1,5,2,3] => [4,5,2,1,3] => [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)
=> ? = 2
[4,1,5,3,2] => [4,3,5,1,2] => [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)
=> ? = 2
[4,2,1,5,3] => [2,4,5,1,3] => [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)
=> ? = 2
[4,2,3,1,5] => [3,4,2,1,5] => [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)
=> ? = 2
[4,2,3,5,1] => [5,4,2,3,1] => [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)
=> ? = 2
[4,2,5,1,3] => [5,4,1,2,3] => [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)
=> ? = 2
[4,2,5,3,1] => [3,5,4,1,2] => [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)
=> ? = 2
[4,3,1,5,2] => [3,4,5,2,1] => [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)
=> ? = 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.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001060The distinguishing index of a graph. St000699The toughness times the least common multiple of 1,. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.