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Matching statistic: St001432
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 2
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000783
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 2
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> ? = 3
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1]
=> ? = 3
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1,1]
=> ? = 3
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2]
=> ? = 3
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [6,3,3]
=> ? = 3
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1]
=> ? = 2
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1]
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [3,6,7,1,2,4,5,8] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,1,1]
=> ? = 3
[[1,2,3,5,8],[4,6,7]]
=> [4,6,7,1,2,3,5,8] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,1]
=> ? = 3
[[1,3,4,6,8],[2,7],[5]]
=> [5,2,7,1,3,4,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2]
=> ? = 3
[[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2]
=> ? = 3
[[1,2,3,6,8],[4,7],[5]]
=> [5,4,7,1,2,3,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2]
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1,1,1]
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1,1]
=> ? = 3
[[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2]
=> ? = 3
[[1,2,4,8],[3,6,7],[5]]
=> [5,3,6,7,1,2,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2]
=> ? = 3
[[1,2,3,8],[4,6,7],[5]]
=> [5,4,6,7,1,2,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2]
=> ? = 3
[[1,4,6,8],[2,5],[3,7]]
=> [3,7,2,5,1,4,6,8] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1]
=> ? = 2
[[1,4,5,8],[2,6],[3,7]]
=> [3,7,2,6,1,4,5,8] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1]
=> ? = 2
[[1,4,8],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2]
=> ? = 3
[[1,3,8],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2]
=> ? = 3
[[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> ? = 2
[[1,4,8],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> ? = 2
[[1,3,8],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> ? = 2
[[1,2,8],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> ? = 2
[[1,2,3,5,6],[4,7,8,9,10]]
=> [4,7,8,9,10,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,1,1,1]
=> ? = 4
[[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,3]
=> ? = 2
[[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1]
=> ? = 3
[[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2]
=> ? = 2
[[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1,1,1]
=> ? = 2
[[1,2,3,4,5,9],[6,7,8]]
=> [6,7,8,1,2,3,4,5,9] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1]
=> ? = 3
[[1,3,4,5,6,7],[2,8,9,10]]
=> [2,8,9,10,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2,1,1,1,1,1,1]
=> ? = 3
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St000662
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 2
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,3,4,5] => ? = 3
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,1,6] => ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => ? = 3
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => ? = 3
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 3
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 2
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 2
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => ? = 3
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,5,3,1,2,6,7] => ? = 3
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,5,3,1,2,6,7] => ? = 3
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 2
[[1,2,6],[3,4,7],[5]]
=> [5,3,4,7,1,2,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 2
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => ? = 3
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => ? = 3
[[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => ? = 3
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 2
[[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 2
[[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,5,1,2,3,6] => ? = 3
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 2
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 2
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 2
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 2
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,5,6,1,7] => ? = 2
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,5,1,6,7] => ? = 2
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => ? = 1
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => ? = 1
[[1,2,4,5,8],[3,6,7]]
=> [3,6,7,1,2,4,5,8] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,5,1,6,7] => ? = 3
[[1,2,3,5,8],[4,6,7]]
=> [4,6,7,1,2,3,5,8] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ? = 3
[[1,3,5,6,7],[2,4,8]]
=> [2,4,8,1,3,5,6,7] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,1,8] => ? = 2
[[1,2,5,6,7],[3,4,8]]
=> [3,4,8,1,2,5,6,7] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,1,7,8] => ? = 2
[[1,3,4,6,7],[2,5,8]]
=> [2,5,8,1,3,4,6,7] => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,1,8] => ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [4,5,8,1,2,3,6,7] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,1,6,7,8] => ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [2,6,8,1,3,4,5,7] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,1,8] => ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [4,6,8,1,2,3,5,7] => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,1,6,7,8] => ? = 2
[[1,2,3,4,7],[5,6,8]]
=> [5,6,8,1,2,3,4,7] => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,1,5,6,7,8] => ? = 2
[[1,3,4,5,6],[2,7,8]]
=> [2,7,8,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,1,8] => ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [3,7,8,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,1,7,8] => ? = 2
[[1,2,3,5,6],[4,7,8]]
=> [4,7,8,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,1,6,7,8] => ? = 2
[[1,2,3,4,6],[5,7,8]]
=> [5,7,8,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,1,5,6,7,8] => ? = 2
[[1,3,4,6,8],[2,7],[5]]
=> [5,2,7,1,3,4,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => ? = 3
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: St000331
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
Description
The number of upper interactions of a Dyck path.
An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St001509
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001509: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
Description
The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly below the diagonal and relative to the trivial lower boundary.
Matching statistic: St000829
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 2
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 2
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,3,4,5] => ? = 3
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ? = 2
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => ? = 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 2
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 2
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 2
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 2
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 2
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,1,6] => ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => ? = 3
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => ? = 3
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => ? = 3
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => ? = 3
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => ? = 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 2
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 3
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 3
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 3
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 2
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 2
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => ? = 2
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St001298
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,2,1,5,6] => 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,4,3,1] => 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,4,2,1] => 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,3,2,1] => 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,1,6] => 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 2
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 2
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,3,2,1,6,7] => ? = 3
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ? = 2
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => ? = 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 2
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 2
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 2
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 2
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 2
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,4,6,5,3,1,7] => ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [3,4,6,5,2,1,7] => ? = 3
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => ? = 3
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => ? = 3
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => ? = 3
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,6,5,3,1] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,6,4,3,1] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,5,4,3,1] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 2
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => ? = 3
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => ? = 3
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => ? = 3
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 2
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 2
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 2
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ? = 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => ? = 2
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => ? = 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => ? = 2
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
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