Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000567
Mapping
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1] => [2] => [2]
 => 0
[2,1] => [1,1] => [2] => [2]
 => 0
[1,1,3] => [2,1] => [1,1] => [1,1]
 => 1
[1,3,1] => [2,1] => [1,1] => [1,1]
 => 1
[3,1,1] => [2,1] => [1,1] => [1,1]
 => 1
[1,2,2] => [1,2] => [1,1] => [1,1]
 => 1
[2,1,2] => [1,2] => [1,1] => [1,1]
 => 1
[2,2,1] => [1,2] => [1,1] => [1,1]
 => 1
[1,2,3] => [1,1,1] => [3] => [3]
 => 0
[1,3,2] => [1,1,1] => [3] => [3]
 => 0
[2,1,3] => [1,1,1] => [3] => [3]
 => 0
[2,3,1] => [1,1,1] => [3] => [3]
 => 0
[3,1,2] => [1,1,1] => [3] => [3]
 => 0
[3,2,1] => [1,1,1] => [3] => [3]
 => 0
[1,1,1,4] => [3,1] => [1,1] => [1,1]
 => 1
[1,1,4,1] => [3,1] => [1,1] => [1,1]
 => 1
[1,4,1,1] => [3,1] => [1,1] => [1,1]
 => 1
[4,1,1,1] => [3,1] => [1,1] => [1,1]
 => 1
[1,1,2,4] => [3,1] => [1,1] => [1,1]
 => 1
[1,1,4,2] => [3,1] => [1,1] => [1,1]
 => 1
[1,2,1,4] => [3,1] => [1,1] => [1,1]
 => 1
[1,2,4,1] => [3,1] => [1,1] => [1,1]
 => 1
[1,4,1,2] => [3,1] => [1,1] => [1,1]
 => 1
[1,4,2,1] => [3,1] => [1,1] => [1,1]
 => 1
[2,1,1,4] => [3,1] => [1,1] => [1,1]
 => 1
[2,1,4,1] => [3,1] => [1,1] => [1,1]
 => 1
[2,4,1,1] => [3,1] => [1,1] => [1,1]
 => 1
[4,1,1,2] => [3,1] => [1,1] => [1,1]
 => 1
[4,1,2,1] => [3,1] => [1,1] => [1,1]
 => 1
[4,2,1,1] => [3,1] => [1,1] => [1,1]
 => 1
[1,1,3,3] => [2,2] => [2] => [2]
 => 0
[1,3,1,3] => [2,2] => [2] => [2]
 => 0
[1,3,3,1] => [2,2] => [2] => [2]
 => 0
[3,1,1,3] => [2,2] => [2] => [2]
 => 0
[3,1,3,1] => [2,2] => [2] => [2]
 => 0
[3,3,1,1] => [2,2] => [2] => [2]
 => 0
[1,1,3,4] => [2,1,1] => [1,2] => [2,1]
 => 2
[1,1,4,3] => [2,1,1] => [1,2] => [2,1]
 => 2
[1,3,1,4] => [2,1,1] => [1,2] => [2,1]
 => 2
[1,3,4,1] => [2,1,1] => [1,2] => [2,1]
 => 2
[1,4,1,3] => [2,1,1] => [1,2] => [2,1]
 => 2
[1,4,3,1] => [2,1,1] => [1,2] => [2,1]
 => 2
[3,1,1,4] => [2,1,1] => [1,2] => [2,1]
 => 2
[3,1,4,1] => [2,1,1] => [1,2] => [2,1]
 => 2
[3,4,1,1] => [2,1,1] => [1,2] => [2,1]
 => 2
[4,1,1,3] => [2,1,1] => [1,2] => [2,1]
 => 2
[4,1,3,1] => [2,1,1] => [1,2] => [2,1]
 => 2
[4,3,1,1] => [2,1,1] => [1,2] => [2,1]
 => 2
[1,2,2,2] => [1,3] => [1,1] => [1,1]
 => 1
[2,1,2,2] => [1,3] => [1,1] => [1,1]
 => 1
Description
The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000454
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 22%
Values
[1,2] => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[2,1,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[2,2,1] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,2,1,2] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,2,2,1] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,2,2,2,2,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,2,2,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,2,2,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,2,1,2,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,2,2,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,2,2,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,2,2,4,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,2,4,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,2,4,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,4,2,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,4,2,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,2,4,4,2,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,4,2,2,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,4,2,2,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,4,2,4,2,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,4,4,2,2,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,2,4,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,4,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,2,4,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,4,2,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,4,2,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,1,4,4,2,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,2,4,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,4,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,1,4,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,2,1,4,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,2,4,1,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,2,4,4,1] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,4,1,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,4,1,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,4,2,1,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,4,2,4,1] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,4,4,1,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,2,4,4,2,1] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,2,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,2,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,4,2,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,2,1,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,2,1,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,2,2,1,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,2,2,4,1] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,2,4,1,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,2,4,2,1] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,4,1,2,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,4,2,1,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,4,2,2,1] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,1,2,2,2,4] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,1,2,2,4,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 22%
Values
[1,2] => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? = 0 + 4
[2,1] => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? = 0 + 4
[1,1,3] => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 4
[1,3,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 4
[3,1,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 4
[1,2,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 4
[2,1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 4
[2,2,1] => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 4
[1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 4
[1,3,2] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 4
[2,1,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 4
[2,3,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 4
[3,1,2] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 4
[3,2,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 4
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 4
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 4
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 4
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 4
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 4
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 4
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 4
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 4
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 4
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 4
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 4
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 4
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 4
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 4
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 0 + 4
[1,1,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,1,1,5,4] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,1,4,1,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,1,4,5,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,1,5,1,4] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,1,5,4,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,4,1,1,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,4,1,5,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,4,5,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,5,1,1,4] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,5,1,4,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,5,4,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[4,1,1,1,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[4,1,1,5,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[4,1,5,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[4,5,1,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[5,1,1,1,4] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[5,1,1,4,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[5,1,4,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[5,4,1,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,2,3,3,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,3,2,3,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,3,3,2,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[1,3,3,3,2] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[2,1,3,3,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
[2,3,1,3,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 2 + 4
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001488
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 22%
Values
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 2 = 0 + 2
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 2 = 0 + 2
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 3 = 1 + 2
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 3 = 1 + 2
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 3 = 1 + 2
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 3 = 1 + 2
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 3 = 1 + 2
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 3 = 1 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2 = 0 + 2
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2 = 0 + 2
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2 = 0 + 2
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2 = 0 + 2
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2 = 0 + 2
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2 = 0 + 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => ? = 1 + 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => ? = 1 + 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => ? = 1 + 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => ? = 1 + 2
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 1 + 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 2
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 2
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 2
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 2 + 2
[1,2,2,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => ? = 1 + 2
[2,1,2,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => ? = 1 + 2
[2,2,1,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => ? = 1 + 2
[2,2,2,1] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => ? = 1 + 2
[1,2,2,3] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[1,2,3,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[1,3,2,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[2,1,2,3] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[2,1,3,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[2,2,1,3] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[2,2,3,1] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[2,3,1,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[2,3,2,1] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[3,1,2,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[3,2,1,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[3,2,2,1] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 1 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[1,2,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[1,3,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[1,3,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[1,4,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[1,4,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[2,1,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[2,1,4,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[2,3,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[2,3,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[2,4,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[2,4,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[3,1,2,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[3,1,4,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[3,2,1,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[3,2,4,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[3,4,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[3,4,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[4,1,2,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[4,1,3,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[4,2,1,3] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[4,2,3,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[4,3,1,2] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
[4,3,2,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2 = 0 + 2
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.