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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St001031
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000319
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [3,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [3,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [3,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,6,8,3] => ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,5,3,8,7] => [1,2,5,6,3,4,8,7] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,7,5,6,3,8] => [1,2,5,6,7,3,4,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,7,5,6,8,3] => [1,2,5,6,8,3,4,7] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,4,8,5,7,6,3] => [1,2,5,7,6,8,3,4] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => [1,2,6,5,7,3,4,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,8,6,5,7,3] => [1,2,6,5,7,8,3,4] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => [1,2,4,5,3,6,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,4,3,6,8,7] => [1,2,4,5,3,6,8,7] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => [1,2,4,7,6,8,3,5] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,6,4,5,3,8,7] => ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,2,6,4,5,7,3,8] => ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => [1,2,4,5,7,6,8,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => [1,2,4,6,5,7,3,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,8,4,6,5,7,3] => ? => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,8,4,7,6,5,3] => [1,2,4,6,7,5,8,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => [1,2,5,4,6,3,8,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,7,5,4,6,8,3] => [1,2,5,4,6,8,3,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => [1,2,6,4,5,7,3,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,5,6,4,8,3,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,8,6,5,4,7,3] => [1,2,5,6,4,7,8,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [1,2,6,5,7,4,8,3] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,6,5,8] => [1,3,2,4,6,7,5,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => [1,3,2,5,4,6,8,7] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,7,4] => [1,3,2,7,8,4,5,6] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,7,5,6,8,4] => [1,3,2,5,6,8,4,7] => ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ? => ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,4,2,3,6,8,5,7] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,7,5,6,2,8] => ? => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,8,5,7,6,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,4,7,6,5,2,8] => ? => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,8,6,7,5,2] => [1,7,5,6,8,2,3,4] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,3,5,4,2,6,8,7] => [1,4,5,2,3,6,8,7] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,4,8,6,7,2] => ? => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,3,6,4,5,2,7,8] => ? => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,6,4,5,7,8,2] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,8,4,5,7,6,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,7,4,6,5,8,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,8,4,6,5,7,2] => [1,4,6,5,7,8,2,3] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,8,4,7,6,5,2] => [1,4,6,7,5,8,2,3] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,5,4,2,8,7] => [1,5,4,6,2,3,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,5,4,7,2,8] => [1,5,4,7,2,3,6,8] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,6,5,4,8,7,2] => [1,5,4,7,8,2,3,6] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,7,5,4,6,2,8] => ? => ?
=> ? = 2
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [3,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [3,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [3,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,6,8,3] => ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,5,3,8,7] => [1,2,5,6,3,4,8,7] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,7,5,6,3,8] => [1,2,5,6,7,3,4,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,7,5,6,8,3] => [1,2,5,6,8,3,4,7] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,4,8,5,7,6,3] => [1,2,5,7,6,8,3,4] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => [1,2,6,5,7,3,4,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,8,6,5,7,3] => [1,2,6,5,7,8,3,4] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => [1,2,4,5,3,6,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,4,3,6,8,7] => [1,2,4,5,3,6,8,7] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => [1,2,4,7,6,8,3,5] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,6,4,5,3,8,7] => ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,2,6,4,5,7,3,8] => ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => [1,2,4,5,7,6,8,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => [1,2,4,6,5,7,3,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,8,4,6,5,7,3] => ? => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,8,4,7,6,5,3] => [1,2,4,6,7,5,8,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => [1,2,5,4,6,3,8,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,7,5,4,6,8,3] => [1,2,5,4,6,8,3,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => [1,2,6,4,5,7,3,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,5,6,4,8,3,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,8,6,5,4,7,3] => [1,2,5,6,4,7,8,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [1,2,6,5,7,4,8,3] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,6,5,8] => [1,3,2,4,6,7,5,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => [1,3,2,5,4,6,8,7] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,7,4] => [1,3,2,7,8,4,5,6] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,7,5,6,8,4] => [1,3,2,5,6,8,4,7] => ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ? => ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,4,2,3,6,8,5,7] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,7,5,6,2,8] => ? => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,8,5,7,6,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,4,7,6,5,2,8] => ? => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,8,6,7,5,2] => [1,7,5,6,8,2,3,4] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,3,5,4,2,6,8,7] => [1,4,5,2,3,6,8,7] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,4,8,6,7,2] => ? => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,3,6,4,5,2,7,8] => ? => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,6,4,5,7,8,2] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,8,4,5,7,6,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,7,4,6,5,8,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,8,4,6,5,7,2] => [1,4,6,5,7,8,2,3] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,8,4,7,6,5,2] => [1,4,6,7,5,8,2,3] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,5,4,2,8,7] => [1,5,4,6,2,3,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,5,4,7,2,8] => [1,5,4,7,2,3,6,8] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ? => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,6,5,4,8,7,2] => [1,5,4,7,8,2,3,6] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,7,5,4,6,2,8] => ? => ?
=> ? = 2
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000147
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [3,1]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [3,1,1]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,6,8,3] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,5,3,8,7] => [1,2,5,6,3,4,8,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,7,5,6,3,8] => [1,2,5,6,7,3,4,8] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,7,5,6,8,3] => [1,2,5,6,8,3,4,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,4,8,5,7,6,3] => [1,2,5,7,6,8,3,4] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => [1,2,6,5,7,3,4,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,8,6,5,7,3] => [1,2,6,5,7,8,3,4] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => [1,2,4,5,3,6,7,8] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,4,3,6,8,7] => [1,2,4,5,3,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => [1,2,4,7,6,8,3,5] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,6,4,5,3,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,2,6,4,5,7,3,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => [1,2,4,5,7,6,8,3] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => [1,2,4,6,5,7,3,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,8,4,6,5,7,3] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,8,4,7,6,5,3] => [1,2,4,6,7,5,8,3] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => [1,2,5,4,6,3,8,7] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,7,5,4,6,8,3] => [1,2,5,4,6,8,3,7] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => [1,2,6,4,5,7,3,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,5,6,4,8,3,7] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,8,6,5,4,7,3] => [1,2,5,6,4,7,8,3] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [1,2,6,5,7,4,8,3] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,6,5,8] => [1,3,2,4,6,7,5,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => [1,3,2,5,4,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,7,4] => [1,3,2,7,8,4,5,6] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,7,5,6,8,4] => [1,3,2,5,6,8,4,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,4,2,3,6,8,5,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,7,5,6,2,8] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,8,5,7,6,2] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,4,7,6,5,2,8] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,8,6,7,5,2] => [1,7,5,6,8,2,3,4] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,3,5,4,2,6,8,7] => [1,4,5,2,3,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,4,8,6,7,2] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,3,6,4,5,2,7,8] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,6,4,5,7,8,2] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,8,4,5,7,6,2] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,7,4,6,5,8,2] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,8,4,6,5,7,2] => [1,4,6,5,7,8,2,3] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,8,4,7,6,5,2] => [1,4,6,7,5,8,2,3] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,5,4,2,8,7] => [1,5,4,6,2,3,8,7] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,5,4,7,2,8] => [1,5,4,7,2,3,6,8] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,6,5,4,8,7,2] => [1,5,4,7,8,2,3,6] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,7,5,4,6,2,8] => ? => ?
=> ? = 2 + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,2]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [2,2]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [3,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [2,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,6,8,5] => [1,2,4,3,6,8,5,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,6,8,3] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,5,3,8,7] => [1,2,5,6,3,4,8,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,7,5,6,3,8] => [1,2,5,6,7,3,4,8] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,7,5,6,8,3] => [1,2,5,6,8,3,4,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,4,8,5,7,6,3] => [1,2,5,7,6,8,3,4] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,2,4,7,6,5,3,8] => [1,2,6,5,7,3,4,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,8,6,5,7,3] => [1,2,6,5,7,8,3,4] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => [1,2,4,5,3,6,7,8] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,4,3,6,8,7] => [1,2,4,5,3,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,2,5,4,7,6,3,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,4,8,6,7,3] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => [1,2,4,7,6,8,3,5] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,6,4,5,3,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,2,6,4,5,7,3,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => [1,2,4,5,7,6,8,3] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,2,7,4,6,5,3,8] => [1,2,4,6,5,7,3,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,8,4,6,5,7,3] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,8,4,7,6,5,3] => [1,2,4,6,7,5,8,3] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => [1,2,5,4,6,3,8,7] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,6,5,4,7,3,8] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,7,5,4,6,8,3] => [1,2,5,4,6,8,3,7] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,7,5,6,4,3,8] => [1,2,6,4,5,7,3,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [1,2,5,6,4,8,3,7] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,8,6,5,4,7,3] => [1,2,5,6,4,7,8,3] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [1,2,6,5,7,4,8,3] => ?
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,6,5,8] => [1,3,2,4,6,7,5,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => [1,3,2,5,4,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,7,4] => [1,3,2,7,8,4,5,6] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,7,5,6,8,4] => [1,3,2,5,6,8,4,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,8,5,7,6,4] => [1,3,2,5,7,6,8,4] => ?
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,6,5,4,8] => [1,3,2,6,5,7,4,8] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,4,2,3,6,8,5,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,4,6,5,7,2,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,7,5,6,2,8] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,8,5,7,6,2] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,4,7,6,5,2,8] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,8,6,7,5,2] => [1,7,5,6,8,2,3,4] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,3,5,4,2,6,8,7] => [1,4,5,2,3,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,4,8,6,7,2] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,3,6,4,5,2,7,8] => ? => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,3,6,4,5,2,8,7] => [1,4,5,6,2,3,8,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,6,4,5,7,8,2] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,8,4,5,7,6,2] => ? => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,7,4,6,5,8,2] => ? => ?
=> ? = 2 + 1
Description
The length of the partition.
Matching statistic: St000254
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => {{1,2,3,4,5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,8,3,4,5,6,7,1] => {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [7,2,3,4,5,6,8,1] => {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,2,3,4,5,6,1,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,8,3,4,5,6,7,2] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,8,4] => {{1,2,3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,7,3,8] => {{1,2},{3,4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,8,4] => {{1},{2,3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,8,4,5,6,7,1] => {{1,3,8},{2},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,6,2,7,8] => {{1},{2,3,4,5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,8,4] => {{1,2},{3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,7,3,8] => {{1},{2},{3,4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,8,4,5,6,7,1] => {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,7,3,4,5,6,8,1] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [2,7,3,4,5,6,1,8] => {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,8,4,5,6,7,2] => {{1},{2,3,8},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [8,2,4,5,6,7,3,1] => {{1,8},{2},{3,4,5,6,7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,2,3,4,5,7,8,1] => {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,2,3,4,5,1,8,7] => {{1,6},{2},{3},{4},{5},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,2,3,4,5,7,1,8] => {{1,6,7},{2},{3},{4},{5},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,2,3,4,5,1,7,8] => {{1,6},{2},{3},{4},{5},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,1,8,4,5,6,7,3] => {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,7,3,4,5,6,8,2] => {{1},{2,7,8},{3},{4},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,8,4,5,6,7,3] => {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,8,4,5,6,7,3,1] => {{1,2,8},{3,4,5,6,7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [8,3,4,5,6,2,7,1] => {{1,8},{2,3,4,5,6},{7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [7,3,4,5,6,2,8,1] => {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1,8] => {{1,7},{2,3,4,5,6},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,8,4,5,6,7,3,2] => {{1},{2,8},{3,4,5,6,7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [8,7,3,4,5,6,2,1] => {{1,8},{2,7},{3},{4},{5},{6}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7,8] => {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,4,1,6,7,8,5] => {{1,2,3,4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,1,5,6,7,4,8] => {{1,2,3},{4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,4,2,6,7,8,5] => {{1},{2,3,4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [4,2,3,8,5,6,7,1] => {{1,4,8},{2},{3},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,3,6,7,8,5] => {{1,2},{3,4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,2,5,6,7,4,8] => {{1},{2,3},{4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => {{1},{2},{3,4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [2,4,3,8,5,6,7,1] => {{1,2,4,8},{3},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,2,7,4,5,6,8,1] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,2,7,4,5,6,1,8] => {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,3,8,5,6,7,2] => {{1},{2,4,8},{3},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [8,3,2,5,6,7,4,1] => {{1,8},{2,3},{4,5,6,7}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2,6,7,8] => {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 1
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000253
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => {{1},{2},{3},{4},{5,6},{7},{8}}
=> {{1},{2},{3},{4},{5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => {{1},{2},{3},{4},{5,6,7},{8}}
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,4,6,8,7,5] => {{1},{2},{3},{4},{5,6,8},{7}}
=> {{1},{2},{3},{4},{5,6,8},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,4,7,6,5,8] => {{1},{2},{3},{4},{5,7},{6},{8}}
=> {{1},{2},{3},{4},{5,7},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => {{1},{2},{3},{4},{5,7,8},{6}}
=> {{1},{2},{3},{4},{5,7,8},{6}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,8,6,7,5] => {{1},{2},{3},{4},{5,8},{6},{7}}
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
=> {{1},{2},{3},{4},{5,7},{6,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3,5,4,7,8,6] => {{1},{2},{3},{4,5},{6,7,8}}
=> {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,7,6] => {{1},{2},{3},{4,5},{6,8},{7}}
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,3,5,6,4,7,8] => {{1},{2},{3},{4,5,6},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,5,6,4,8,7] => {{1},{2},{3},{4,5,6},{7,8}}
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,5,6,7,4,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,3,5,6,8,7,4] => {{1},{2},{3},{4,5,6,8},{7}}
=> {{1},{2},{3},{4,5,6,8},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,3,5,7,6,4,8] => {{1},{2},{3},{4,5,7},{6},{8}}
=> {{1},{2},{3},{4,5,7},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,6,8,4] => {{1},{2},{3},{4,5,7,8},{6}}
=> {{1},{2},{3},{4,5,7,8},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,8,6,7,4] => {{1},{2},{3},{4,5,8},{6},{7}}
=> {{1},{2},{3},{4,5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,7,6,4] => {{1},{2},{3},{4,5,8},{6,7}}
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8] => {{1},{2},{3},{4,6},{5},{7},{8}}
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,5,4,8,7] => {{1},{2},{3},{4,6},{5},{7,8}}
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,5,7,4,8] => {{1},{2},{3},{4,6,7},{5},{8}}
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => {{1},{2},{3},{4,6,7,8},{5}}
=> {{1},{2},{3},{4,6,7,8},{5}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,5,8,7,4] => {{1},{2},{3},{4,6,8},{5},{7}}
=> {{1},{2},{3},{4,6,8},{5},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,7,5,6,4,8] => {{1},{2},{3},{4,7},{5},{6},{8}}
=> {{1},{2},{3},{4,7},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,7,5,6,8,4] => {{1},{2},{3},{4,7,8},{5},{6}}
=> {{1},{2},{3},{4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,8,5,6,7,4] => {{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,8,5,7,6,4] => {{1},{2},{3},{4,8},{5},{6,7}}
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,6,5,4,8] => {{1},{2},{3},{4,7},{5,6},{8}}
=> {{1},{2},{3},{4,6},{5,7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => {{1},{2},{3},{4,7,8},{5,6}}
=> {{1},{2},{3},{4,6},{5,7,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,8,6,5,7,4] => {{1},{2},{3},{4,8},{5,6},{7}}
=> {{1},{2},{3},{4,6},{5,8},{7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,8,6,7,5,4] => {{1},{2},{3},{4,8},{5,6,7}}
=> {{1},{2},{3},{4,6,8},{5,7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => {{1},{2},{3},{4,8},{5,7},{6}}
=> {{1},{2},{3},{4,7},{5,8},{6}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8] => {{1},{2},{3,4},{5},{6},{7},{8}}
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => {{1},{2},{3,4},{5},{6,7},{8}}
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,4,3,5,7,8,6] => {{1},{2},{3,4},{5},{6,7,8}}
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,3,5,8,7,6] => {{1},{2},{3,4},{5},{6,8},{7}}
=> {{1},{2},{3,4},{5},{6,8},{7}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => {{1},{2},{3,4},{5,6},{7,8}}
=> {{1},{2},{3,4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,4,3,6,7,5,8] => {{1},{2},{3,4},{5,6,7},{8}}
=> ?
=> ? = 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000097
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,5,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,3,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => [2,1,6,7,3,4,5] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => [2,1,5,7,3,4,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,7,5,6,3] => [2,1,5,6,7,3,4] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [2,1,6,5,7,3,4] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,4,6,7,3] => [2,1,4,7,3,5,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,4,7,6,3] => [2,1,4,6,7,3,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,6,4,5,7,3] => [2,1,4,5,7,3,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,7,4,6,5,3] => [2,1,4,6,5,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => [2,1,5,4,7,3,6] => ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,7,5,4,6,3] => [2,1,5,4,6,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => [2,1,6,4,5,7,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => [3,1,2,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,5,7,4] => [3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,7,5,6,4] => [3,1,2,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [3,1,2,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [4,1,2,3,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [6,5,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,5,4,1,7,6] => [4,5,1,2,3,7,6] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => [4,7,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => [4,6,7,1,2,3,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,6,4,5,7,1] => [4,5,7,1,2,3,6] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => [4,6,5,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [5,4,7,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,7,5,4,6,1] => [5,4,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => [6,4,5,7,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => [5,6,4,7,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => [3,4,1,2,7,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => [3,4,1,2,6,7,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => [3,5,1,2,4,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => [3,7,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => [3,6,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,7,5,6,1] => [3,5,6,7,1,2,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [3,6,5,7,1,2,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,3,4,1,7,6] => [3,4,5,1,2,7,6] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,3,4,6,7,1] => [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,3,4,7,6,1] => [3,4,6,7,1,2,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St001277
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,5,7,6,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,5,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,2,3,4,6,8,5,7] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,2,3,4,7,5,6,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,2,3,4,7,5,8,6] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,2,3,4,7,8,5,6] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,7,8,6] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,2,3,5,4,8,6,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,3,5,6,4,7,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,6,4,8,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,6,7,4,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,8,4,7] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,7,4,8,6] => ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,2,3,5,8,4,6,7] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,7,8,4,6] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,8,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,2,3,6,4,7,8,5] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,2,3,6,4,8,5,7] => ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,2,3,7,4,5,8,6] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,2,3,7,4,8,5,6] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,2,3,6,7,4,5,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,2,3,6,7,4,8,5] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,2,3,6,8,4,5,7] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,7,8,4,5] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,2,3,7,8,4,5,6] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,3,5,7,8,6] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,4,3,5,8,6,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,8,7] => ([(2,7),(3,6),(4,5)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,6,7,5,8] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
Description
The degeneracy of a graph.
The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
Matching statistic: St001358
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,5,7,6,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,5,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,2,3,4,6,8,5,7] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,2,3,4,7,5,6,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,2,3,4,7,5,8,6] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,2,3,4,7,8,5,6] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,3,5,4,6,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,7,8,6] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,2,3,5,4,8,6,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,3,5,6,4,7,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,6,4,8,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,6,7,4,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,8,4,7] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,7,4,8,6] => ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,2,3,5,8,4,6,7] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,7,8,4,6] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,8,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,2,3,6,4,7,8,5] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,2,3,6,4,8,5,7] => ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,2,3,7,4,5,8,6] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,2,3,7,4,8,5,6] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,2,3,6,7,4,5,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,2,3,6,7,4,8,5] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,2,3,6,8,4,5,7] => ([(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,7,8,4,5] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,2,3,7,8,4,5,6] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,3,5,7,8,6] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,4,3,5,8,6,7] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,8,7] => ([(2,7),(3,6),(4,5)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,6,7,5,8] => ([(3,4),(5,7),(6,7)],8)
=> ? = 1
Description
The largest degree of a regular subgraph of a graph.
For $k > 2$, it is an NP-complete problem to determine whether a graph has a $k$-regular subgraph, see [1].
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001029The size of the core of a graph. St000098The chromatic number of a graph. St000527The width of the poset. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000929The constant term of the character polynomial of an integer partition. St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000387The matching number of a graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001568The smallest positive integer that does not appear twice in the partition. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000914The sum of the values of the Möbius function of a poset. St001427The number of descents of a signed permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001890The maximum magnitude of the Möbius function of a poset. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice. St000632The jump number of the poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St001960The number of descents of a permutation minus one if its first entry is not one. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000640The rank of the largest boolean interval in a poset.
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