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Your data matches 245 different statistics following compositions of up to 3 maps.
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Matching statistic: St001505
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(load all 10 compositions to match this statistic)
St001505: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
Description
The number of elements generated by the Dyck path as a map in the full transformation monoid.
We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.
Matching statistic: St000325
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,1,2,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,4,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => [4,2,1,3] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [4,2,3,1] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [1,3,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => [1,3,2,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => [5,1,2,4,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => [1,5,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => [5,4,1,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => [1,5,3,4,2] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => [1,3,2,5,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => [2,1,5,4,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => [3,1,5,4,2] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => [3,5,4,2,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => [2,5,4,1,3] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => [5,4,2,1,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => [5,4,2,3,1] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [2,5,3,4,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => [5,1,3,4,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => [1,2,4,5,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => [1,2,4,3,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => [1,4,3,5,2] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => [1,4,3,2,5] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => 3
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,1,2,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,4,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => [4,2,1,3] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [4,2,3,1] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [1,3,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => [1,3,2,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => [5,1,2,4,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => [1,5,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => [5,4,1,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => [1,5,3,4,2] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => [1,3,2,5,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => [2,1,5,4,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => [3,1,5,4,2] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => [3,5,4,2,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => [2,5,4,1,3] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => [5,4,2,1,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => [5,4,2,3,1] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [2,5,3,4,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => [5,1,3,4,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => [1,2,4,5,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => [1,2,4,3,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => [1,4,3,5,2] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => [1,4,3,2,5] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => 3
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,1,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => [4,2,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [4,2,3,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => [5,1,2,4,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => [1,5,2,4,3] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => [5,4,1,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => [1,5,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => [1,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => [3,1,5,4,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => [3,5,4,2,1] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => [2,1,5,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => [5,4,2,1,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => [5,4,2,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [2,5,3,4,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => [5,1,3,4,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => [1,4,3,5,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => [1,4,3,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => 2 = 3 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000374
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [3,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,4,1,2] => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [4,1,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [1,3,4,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[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 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,5,1,2,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,5,1,3] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => [1,3,4,2,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => [1,3,5,2,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => [2,4,1,5,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [3,1,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,4,5,2,3] => [1,4,2,5,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [1,5,2,3,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => [4,1,2,5,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => [1,5,3,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [4,1,5,2,3] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => [3,1,4,5,2] => 2 = 3 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000662
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,2,1] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [4,3,1,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,4,2,1] => [4,1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [2,3,4,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,1,3,2] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2 = 3 - 1
[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 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [5,4,1,2,3] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [4,2,5,1,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [5,1,3,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [4,3,1,2,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [5,4,3,1,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,5,2,4,1] => [4,5,1,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,4,2,1,5] => [4,1,3,2,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,5,2,1,4] => [5,4,1,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,5,3,2,1] => [3,5,1,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => [2,5,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [2,4,1,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,3,5,2,1] => [5,1,4,2,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,3,1,4] => [2,3,5,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,3,4,2,1] => [4,2,3,5,1] => 2 = 3 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000996
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 1 = 2 - 1
[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 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [3,4,2,1,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [4,5,3,2,1] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [3,4,1,5,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => [3,1,5,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [3,2,5,4,1] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,4,1,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,3,5,2,1] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [3,2,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [4,3,1,5,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [3,1,4,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [4,1,5,3,2] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [4,2,5,3,1] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [3,1,2,5,4] => 2 = 3 - 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001812
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],2)
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,6} - 1
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Matching statistic: St001644
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5} - 1
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000288
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [[1],[2]]
=> 1 => 1 => 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> 101 => 011 => 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> 010 => 100 => 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 10101 => 01011 => 3
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 10010 => 00101 => 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 01001 => 10010 => 2
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 01010 => 10100 => 2
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 00100 => 01000 => 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 1010101 => 0101011 => 4
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 1010010 => 0100101 => 3
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 1001001 => 0010011 => 3
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 1001010 => 0010101 => 3
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 1000100 => 0001001 => 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0100101 => 1001010 => 3
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0100010 => 1000100 => 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0101001 => 1010010 => 3
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 0101010 => 1010100 => 3
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 0100100 => 1001000 => 2
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 0010001 => 0100010 => 2
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0010010 => 0100100 => 2
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 0010100 => 0101000 => 2
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 0001000 => 0010000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 101010101 => 010101011 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 101010010 => 010100101 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 101001001 => 010010011 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 101001010 => 010010101 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 101000100 => 010001001 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 100100101 => 001001011 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 100100010 => 001000101 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 100101001 => 001010011 => ? ∊ {2,3,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 100101010 => 001010101 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 100100100 => 001001001 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 100010001 => 000100011 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 100010010 => 000100101 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 100010100 => 000101001 => ? ∊ {2,3,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 100001000 => 000010001 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 010010101 => 100101010 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 010010010 => 100100100 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 010001001 => 100010010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 010001010 => 100010100 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 010000100 => 100001000 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 010100101 => 101001010 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 010100010 => 101000100 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 010101001 => 101010010 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 010101010 => 101010100 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> 010100100 => 101001000 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 010010001 => 100100010 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> 010010010 => 100100100 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 010010100 => 100101000 => ? ∊ {2,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> 010001000 => 100010000 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 001000101 => 010001010 => 3
[1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 001000010 => 010000100 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 001001001 => 010010010 => 3
[1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> 000101000 => 001010000 => ? ∊ {2,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> 10101010101 => 01010101011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> 10101001001 => 01010010011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> 10100100101 => 01001001011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> 10100101001 => 01001010011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> 10100010001 => 01000100011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,3,4,7,9,11],[2,5,6,8,10,12]]
=> 10010010101 => 00100101011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,4,7,8,11],[2,5,6,9,10,12]]
=> 10010001001 => 00100010011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,3,4,6,9,11],[2,5,7,8,10,12]]
=> 10010100101 => 00101001011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,3,4,6,8,11],[2,5,7,9,10,12]]
=> 10010101001 => 00101010011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> 10010101010 => 00101010101 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,3,4,6,7,11],[2,5,8,9,10,12]]
=> 10010010001 => 00100100011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> 10001000101 => 00010001011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,3,4,5,8,11],[2,6,7,9,10,12]]
=> 10001001001 => 00010010011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> 10001001010 => 00010010101 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> 10001010001 => 00010100011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> 10001010010 => 00010100101 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> 10001010100 => 00010101001 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,4,5,6,11],[2,7,8,9,10,12]]
=> 10000100001 => 00001000011 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> 10000100010 => 00001000101 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> 10000100100 => 00001001001 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> 10000101000 => 00001010001 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> 10000010000 => 00000100001 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> 01010101010 => 10101010100 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,8,9],[3,5,7,10,11,12]]
=> 01010100100 => 10101001000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,6,7,9],[3,5,8,10,11,12]]
=> 01010010100 => 10100101000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,6,7,8],[3,5,9,10,11,12]]
=> 01010001000 => 10100010000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,2,4,5,7,9],[3,6,8,10,11,12]]
=> 01001010100 => 10010101000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,2,4,5,7,8],[3,6,9,10,11,12]]
=> 01001001000 => 10010010000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [[1,2,4,5,6,8],[3,7,9,10,11,12]]
=> 01000101000 => 10001010000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,2,4,5,6,7],[3,8,9,10,11,12]]
=> 01000010000 => 10000100000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> 00101010100 => 01010101000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,7,8],[4,6,9,10,11,12]]
=> 00101001000 => 01010010000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [[1,2,3,5,6,8],[4,7,9,10,11,12]]
=> 00100101000 => 01001010000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,2,3,5,6,7],[4,8,9,10,11,12]]
=> 00100010000 => 01000100000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [[1,2,3,4,6,8],[5,7,9,10,11,12]]
=> 00010101000 => 00101010000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,2,3,4,6,7],[5,8,9,10,11,12]]
=> 00010010000 => 00100100000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,2,3,4,5,7],[6,8,9,10,11,12]]
=> 00001010000 => 00010100000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> 00000100000 => 00001000000 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6}
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 235 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St000703The number of deficiencies of a permutation. St000291The number of descents of a binary word. St000742The number of big ascents of a permutation after prepending zero. St000741The Colin de Verdière graph invariant. St001427The number of descents of a signed permutation. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St001432The order dimension of the partition. St001489The maximum of the number of descents and the number of inverse descents. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000157The number of descents of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000259The diameter of a connected graph. St001566The length of the longest arithmetic progression in a permutation. St000264The girth of a graph, which is not a tree. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000292The number of ascents of a binary word. St000831The number of indices that are either descents or recoils. St000646The number of big ascents of a permutation. St000260The radius of a connected graph. St001645The pebbling number of a connected graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000390The number of runs of ones in a binary word. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001875The number of simple modules with projective dimension at most 1. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000822The Hadwiger number of the graph. St000028The number of stack-sorts needed to sort a permutation. St000711The number of big exceedences of a permutation. St000308The height of the tree associated to a permutation. St000245The number of ascents of a permutation. St000647The number of big descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000054The first entry of the permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001060The distinguishing index of a graph. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). 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 $eA$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000702The number of weak deficiencies of a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000710The number of big deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001120The length of a longest path in a graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001298The number of repeated entries in the Lehmer code of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001093The detour number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary 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 $eA$. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000474Dyson's crank of a partition. St001175The size of a partition minus the hook length of the base cell. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001488The number of corners of a skew partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000389The number of runs of ones of odd length in a binary word. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000455The second largest eigenvalue of a graph if it is integral. St000837The number of ascents of distance 2 of a permutation. St000358The number of occurrences of the pattern 31-2. St000836The number of descents of distance 2 of a permutation. St001864The number of excedances of a signed permutation. St000507The number of ascents of a standard tableau. St000746The number of pairs with odd minimum in a perfect matching. St001866The nesting alignments of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001083The number of boxed occurrences of 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000013The height of a Dyck path. St000007The number of saliances of the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000451The length of the longest pattern of the form k 1 2. St000884The number of isolated descents of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000829The Ulam distance of a permutation to the identity permutation. St001061The number of indices that are both descents and recoils of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000201The number of leaf nodes in a binary tree. St000214The number of adjacencies of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000445The number of rises of length 1 of a Dyck path. St000648The number of 2-excedences of a permutation. St001115The number of even descents of a permutation. St001153The number of blocks with even minimum in a set partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St000015The number of peaks of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000942The number of critical left to right maxima of the parking functions. St000991The number of right-to-left minima of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000120The number of left tunnels of a Dyck path. St000808The number of up steps of the associated bargraph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000164The number of short pairs. St000216The absolute length of a permutation. St000239The number of small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000809The reduced reflection length of the permutation. St000912The number of maximal antichains in a poset. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001343The dimension of the reduced incidence algebra of a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000053The number of valleys of the Dyck path. St000097The order of the largest clique of the graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000236The number of cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000676The number of odd rises of a Dyck path. St000872The number of very big descents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001267The length of the Lyndon factorization of the binary word. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001649The length of a longest trail in a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001874Lusztig's a-function for the symmetric group. St001964The interval resolution global dimension of a poset. St000024The number of double up and double down steps of a Dyck path. St000144The pyramid weight of the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000392The length of the longest run of ones in a binary word. St000381The largest part of an integer composition. St001581The achromatic number of a graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000891The number of distinct diagonal sums of a permutation matrix. St001435The number of missing boxes in the first row. St001877Number of indecomposable injective modules with projective dimension 2. St000237The number of small exceedances. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation.
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