Your data matches 179 different statistics following compositions of up to 3 maps.
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St000686: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The finitistic dominant dimension of a Dyck path. To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
St001294: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]]. The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
St000308: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => 2
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,1,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,1,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,5,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => 2
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00071: Permutations descent compositionInteger compositions
St000381: Integer compositions ⟶ 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]
=> [1,2] => [1,2] => [2] => 2
[1,1,0,0]
=> [2,1] => [2,1] => [1,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,2] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,3] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => [1,2,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => [1,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [1,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [2,1,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,4] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => [1,3,1] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [1,1,3] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [1,2,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [1,3,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,4,2] => [1,1,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,1,5,2] => [1,1,2,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [5,1,3,4,2] => [1,3,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,4] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [1,1,3] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [1,1,3] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [1,1,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,4,2,1,3] => [1,1,1,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [1,2,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,3,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,2,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,3,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,3,4,1] => [2,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,2,1] => 2
Description
The largest part of an integer composition.
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00239: Permutations CorteelPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000028: 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] => [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] => [3,2,1] => [2,3,1] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [3,2,1] => 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] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 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] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [4,3,2,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => [4,1,3,2] => 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] => [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => [1,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => [1,4,5,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => 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] => [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => [2,3,5,4,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => [2,4,3,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => [2,4,5,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => [2,5,1,4,3] => 2 = 3 - 1
Description
The number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
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] => [2,3,1] => [3,2,1] => 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] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [2,3,1] => 2 = 3 - 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] => [4,2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,4,2,1] => [2,4,3,1] => 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] => [4,3,2,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,1,3,2] => [4,3,1,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 3 = 4 - 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] => [5,2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [5,2,3,1,4] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [5,4,3,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [5,2,1,3,4] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,5,2,4,1] => [4,5,3,2,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,4,2,1,5] => [2,4,3,1,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,5,2,1,4] => [2,5,3,1,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,1,4,2] => [5,4,3,1,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,5,3,2,1] => [2,3,5,4,1] => 3 = 4 - 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] => [5,3,2,4,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [5,3,2,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,3,5,2,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [5,4,2,3,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [4,5,2,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,4,2,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,3,1,4] => [3,5,2,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,2,1,4,3] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,3,4,2,1] => [2,4,5,3,1] => 3 = 4 - 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.
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
St000485: Permutations ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 3
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [3,4,2,1] => 4
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [4,1,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,4,3,1] => [3,2,4,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 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] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,4,5,3,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [1,5,4,3,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => [1,5,2,4,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,4,2] => [1,4,3,5,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[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] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [3,4,2,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [4,5,2,3,1] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [5,4,2,3,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [5,4,1,2,3] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [4,3,2,5,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [5,3,2,4,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 2
Description
The length of the longest cycle of a permutation.
Matching statistic: St001062
Mp00138: Dyck paths to noncrossing partitionSet partitions
Mp00216: Set partitions inverse Wachs-WhiteSet partitions
Mp00220: Set partitions YipSet partitions
St001062: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> {{1}}
=> {{1}}
=> ? = 1
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> 3
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4
[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}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> {{1},{2,3,5},{4}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,4},{3,5}}
=> {{1,5},{2,4},{3}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> {{1,5},{2},{3,4}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3
Description
The maximal size of a block of a set partition.
Matching statistic: St000326
Mp00093: Dyck paths to binary wordBinary words
Mp00105: Binary words complementBinary words
Mp00224: Binary words runsortBinary words
St000326: Binary words ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 01 => 01 => 2 = 1 + 1
[1,0,1,0]
=> 1010 => 0101 => 0101 => 2 = 1 + 1
[1,1,0,0]
=> 1100 => 0011 => 0011 => 3 = 2 + 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 010101 => 2 = 1 + 1
[1,0,1,1,0,0]
=> 101100 => 010011 => 001101 => 3 = 2 + 1
[1,1,0,0,1,0]
=> 110010 => 001101 => 001101 => 3 = 2 + 1
[1,1,0,1,0,0]
=> 110100 => 001011 => 001011 => 3 = 2 + 1
[1,1,1,0,0,0]
=> 111000 => 000111 => 000111 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 01010101 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01010011 => 00110101 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 01001101 => 00110101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 01001011 => 00101011 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 01000111 => 00011101 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 00110101 => 00110101 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 00110011 => 00110011 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 00101101 => 00101011 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 00101011 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 00100111 => 00100111 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 00011101 => 00011101 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 00011011 => 00011011 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 00010111 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 00001111 => 00001111 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 0101010101 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101010011 => 0011010101 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101001101 => 0011010101 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101001011 => 0010101011 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101000111 => 0001110101 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0100110101 => 0011010101 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0100110011 => 0011001101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0100101101 => 0010101011 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0100101011 => 0010101011 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0100100111 => 0010011101 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0100011101 => 0001110101 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0100011011 => 0001101011 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0100010111 => 0001010111 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0100001111 => 0000111101 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0011010101 => 0011010101 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0011010011 => 0011001101 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0011001101 => 0011001101 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0011001011 => 0010011011 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0011000111 => 0001110011 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0010110101 => 0010101011 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0010110011 => 0010011011 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0010101101 => 0010101011 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0010101011 => 0010101011 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0010100111 => 0010011101 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0010011101 => 0010011101 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0010011011 => 0010011011 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0010010111 => 0010010111 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0010001111 => 0001111001 => 4 = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 010010001111 => 000111100101 => ? ∊ {3,3,4,4} + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> 110101110000 => 001010001111 => 000111100101 => ? ∊ {3,3,4,4} + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 001000111101 => 000111100101 => ? ∊ {3,3,4,4} + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> 110111100000 => 001000011111 => 000011111001 => ? ∊ {3,3,4,4} + 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
The following 169 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000983The length of the longest alternating subword. St001933The largest multiplicity of a part in an integer partition. St000993The multiplicity of the largest part of an integer partition. St001372The length of a longest cyclic run of ones of a binary word. St000982The length of the longest constant subword. St000392The length of the longest run of ones in a binary word. St000144The pyramid weight of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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 $eA$. St000271The chromatic index of a graph. St001638The book thickness of a graph. St000444The length of the maximal rise of a Dyck path. St001330The hat guessing number of a graph. St000366The number of double descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001060The distinguishing index of a graph. 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 $eA$. St001644The dimension of a graph. St000732The number of double deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000141The maximum drop size of a permutation. St000451The length of the longest pattern of the form k 1 2. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000893The number of distinct diagonal sums of an alternating sign matrix. St001566The length of the longest arithmetic progression in a permutation. St000731The number of double exceedences of a permutation. St000264The girth of a graph, which is not a tree. St000007The number of saliances of the permutation. St000374The number of exclusive right-to-left minima of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000328The maximum number of child nodes in a tree. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000778The metric dimension of a graph. St000846The maximal number of elements covering an element of a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001530The depth of a Dyck path. St001589The nesting number of a perfect matching. St000155The number of exceedances (also excedences) of a permutation. St000171The degree of the graph. St000245The number of ascents of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001118The acyclic chromatic index of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000528The height of a poset. St000907The number of maximal antichains of minimal length in 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. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000845The maximal number of elements covered by an element in a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001545The second Elser number of a connected graph. St000474Dyson's crank of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001432The order dimension of the partition. St001488The number of corners of a skew partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001864The number of excedances of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000662The staircase size of the code 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001730The number of times the path corresponding to a binary word crosses the base line. St000120The number of left tunnels of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000670The reversal length of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001742The difference of the maximal and the minimal degree in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix 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. St001557The number of inversions of the second entry of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001624The breadth of a lattice. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001435The number of missing boxes in the first row. St001877Number of indecomposable injective modules with projective dimension 2. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000703The number of deficiencies of a permutation. St000022The number of fixed points of a permutation. St000390The number of runs of ones in a binary word.