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Matching statistic: St000459
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,2] => [2]
=> 2
[2,1] => [1,1]
=> 2
[1,2,3] => [3]
=> 3
[1,3,2] => [2,1]
=> 3
[2,1,3] => [2,1]
=> 3
[2,3,1] => [2,1]
=> 3
[3,1,2] => [2,1]
=> 3
[3,2,1] => [1,1,1]
=> 3
[1,2,3,4] => [4]
=> 4
[1,2,4,3] => [3,1]
=> 4
[1,3,2,4] => [3,1]
=> 4
[1,3,4,2] => [3,1]
=> 4
[1,4,2,3] => [3,1]
=> 4
[1,4,3,2] => [2,1,1]
=> 4
[2,1,3,4] => [3,1]
=> 4
[2,1,4,3] => [2,2]
=> 3
[2,3,1,4] => [3,1]
=> 4
[2,3,4,1] => [3,1]
=> 4
[2,4,1,3] => [2,2]
=> 3
[2,4,3,1] => [2,1,1]
=> 4
[3,1,2,4] => [3,1]
=> 4
[3,1,4,2] => [2,2]
=> 3
[3,2,1,4] => [2,1,1]
=> 4
[3,2,4,1] => [2,1,1]
=> 4
[3,4,1,2] => [2,2]
=> 3
[3,4,2,1] => [2,1,1]
=> 4
[4,1,2,3] => [3,1]
=> 4
[4,1,3,2] => [2,1,1]
=> 4
[4,2,1,3] => [2,1,1]
=> 4
[4,2,3,1] => [2,1,1]
=> 4
[4,3,1,2] => [2,1,1]
=> 4
[4,3,2,1] => [1,1,1,1]
=> 4
[1,2,3,4,5] => [5]
=> 5
[1,2,3,5,4] => [4,1]
=> 5
[1,2,4,3,5] => [4,1]
=> 5
[1,2,4,5,3] => [4,1]
=> 5
[1,2,5,3,4] => [4,1]
=> 5
[1,2,5,4,3] => [3,1,1]
=> 5
[1,3,2,4,5] => [4,1]
=> 5
[1,3,2,5,4] => [3,2]
=> 4
[1,3,4,2,5] => [4,1]
=> 5
[1,3,4,5,2] => [4,1]
=> 5
[1,3,5,2,4] => [3,2]
=> 4
[1,3,5,4,2] => [3,1,1]
=> 5
[1,4,2,3,5] => [4,1]
=> 5
[1,4,2,5,3] => [3,2]
=> 4
[1,4,3,2,5] => [3,1,1]
=> 5
[1,4,3,5,2] => [3,1,1]
=> 5
[1,4,5,2,3] => [3,2]
=> 4
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000203
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [.,.]
=> 1
[1,2] => [[1,2]]
=> [1,2] => [.,[.,.]]
=> 2
[2,1] => [[1],[2]]
=> [2,1] => [[.,.],.]
=> 2
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> 3
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 3
[2,3,1] => [[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 3
[3,1,2] => [[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> 3
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> 3
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 4
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 4
[1,3,4,2] => [[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 4
[1,4,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 4
[1,4,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 4
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4
[2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 3
[2,3,1,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4
[2,3,4,1] => [[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4
[2,4,1,3] => [[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 3
[2,4,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 4
[3,1,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 4
[3,1,4,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 4
[3,2,4,1] => [[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 4
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3
[3,4,2,1] => [[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 4
[4,1,2,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 4
[4,1,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 4
[4,2,1,3] => [[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 4
[4,2,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 4
[4,3,1,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 4
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 4
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> 4
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> 4
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 4
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 4
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000395
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001004
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,2] => 2
[2,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 2
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 3
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 3
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 3
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 3
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 4
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 4
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001020
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001348
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St000543
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 01 => 2 = 1 + 1
[1,2] => [2]
=> 100 => 001 => 3 = 2 + 1
[2,1] => [1,1]
=> 110 => 011 => 3 = 2 + 1
[1,2,3] => [3]
=> 1000 => 0001 => 4 = 3 + 1
[1,3,2] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[2,1,3] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[2,3,1] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[3,1,2] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[3,2,1] => [1,1,1]
=> 1110 => 0111 => 4 = 3 + 1
[1,2,3,4] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[1,2,4,3] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,3,2,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,3,4,2] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,4,2,3] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,4,3,2] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[2,1,3,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[2,1,4,3] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[2,3,1,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[2,3,4,1] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[2,4,1,3] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[2,4,3,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[3,1,2,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[3,1,4,2] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[3,2,4,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[3,4,1,2] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[3,4,2,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,1,2,3] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[4,1,3,2] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,2,1,3] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,2,3,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,3,1,2] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,3,2,1] => [1,1,1,1]
=> 11110 => 01111 => 5 = 4 + 1
[1,2,3,4,5] => [5]
=> 100000 => 000001 => 6 = 5 + 1
[1,2,3,5,4] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,4,3,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,4,5,3] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,5,3,4] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,5,4,3] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,3,2,4,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,3,2,5,4] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
[1,3,4,2,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,3,4,5,2] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,3,5,2,4] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,4,2,3,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,4,2,5,3] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
[1,4,3,2,5] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,4,5,2,3] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
Description
The size of the conjugacy class of a binary word.
Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
Matching statistic: St000626
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000626: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 01 => 2 = 1 + 1
[1,2] => [2]
=> 100 => 001 => 3 = 2 + 1
[2,1] => [1,1]
=> 110 => 011 => 3 = 2 + 1
[1,2,3] => [3]
=> 1000 => 0001 => 4 = 3 + 1
[1,3,2] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[2,1,3] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[2,3,1] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[3,1,2] => [2,1]
=> 1010 => 0011 => 4 = 3 + 1
[3,2,1] => [1,1,1]
=> 1110 => 0111 => 4 = 3 + 1
[1,2,3,4] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[1,2,4,3] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,3,2,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,3,4,2] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,4,2,3] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[1,4,3,2] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[2,1,3,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[2,1,4,3] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[2,3,1,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[2,3,4,1] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[2,4,1,3] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[2,4,3,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[3,1,2,4] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[3,1,4,2] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[3,2,4,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[3,4,1,2] => [2,2]
=> 1100 => 0011 => 4 = 3 + 1
[3,4,2,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,1,2,3] => [3,1]
=> 10010 => 00011 => 5 = 4 + 1
[4,1,3,2] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,2,1,3] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,2,3,1] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,3,1,2] => [2,1,1]
=> 10110 => 00111 => 5 = 4 + 1
[4,3,2,1] => [1,1,1,1]
=> 11110 => 01111 => 5 = 4 + 1
[1,2,3,4,5] => [5]
=> 100000 => 000001 => 6 = 5 + 1
[1,2,3,5,4] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,4,3,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,4,5,3] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,5,3,4] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,2,5,4,3] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,3,2,4,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,3,2,5,4] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
[1,3,4,2,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,3,4,5,2] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,3,5,2,4] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,4,2,3,5] => [4,1]
=> 100010 => 000011 => 6 = 5 + 1
[1,4,2,5,3] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
[1,4,3,2,5] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => 000111 => 6 = 5 + 1
[1,4,5,2,3] => [3,2]
=> 10100 => 00011 => 5 = 4 + 1
Description
The minimal period of a binary word.
This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.
Matching statistic: St000998
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[2,1] => [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000806
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => ([],1)
=> [1] => ? = 1 + 1
[1,2] => ([],2)
=> [2] => 3 = 2 + 1
[2,1] => ([(0,1)],2)
=> [1,1] => 3 = 2 + 1
[1,2,3] => ([],3)
=> [3] => 4 = 3 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1] => 4 = 3 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1] => 4 = 3 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [2,1] => 4 = 3 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [2,1] => 4 = 3 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 4 = 3 + 1
[1,2,3,4] => ([],4)
=> [4] => 5 = 4 + 1
[1,2,4,3] => ([(2,3)],4)
=> [3,1] => 5 = 4 + 1
[1,3,2,4] => ([(2,3)],4)
=> [3,1] => 5 = 4 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1] => 5 = 4 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1] => 5 = 4 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[2,1,3,4] => ([(2,3)],4)
=> [3,1] => 5 = 4 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => 4 = 3 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1] => 5 = 4 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 5 = 4 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [2,2] => 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1] => 5 = 4 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [2,2] => 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 5 = 4 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 5 = 4 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 5 = 4 + 1
[1,2,3,4,5] => ([],5)
=> [5] => 6 = 5 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 6 = 5 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [3,2] => 5 = 4 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [3,2] => 5 = 4 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 6 = 5 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [4,1] => 6 = 5 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [3,2] => 5 = 4 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 6 = 5 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 6 = 5 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 5 = 4 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 6 = 5 + 1
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001267The length of the Lyndon factorization of the binary word. St000144The pyramid weight of the Dyck path. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000026The position of the first return of a Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000924The number of topologically connected components of a perfect matching. St000019The cardinality of the support of a permutation. St001480The number of simple summands of the module J^2/J^3. St001958The degree of the polynomial interpolating the values of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001468The smallest fixpoint of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000863The length of the first row of the shifted shape of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000080The rank of the poset. St000454The largest eigenvalue of a graph if it is integral. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000528The height of a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001782The order of rowmotion on the set of order ideals of a poset. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000097The order of the largest clique of the graph. St001720The minimal length of a chain of small intervals in a lattice. St001792The arboricity of a graph. St001581The achromatic number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001621The number of atoms of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000299The number of nonisomorphic vertex-induced subtrees. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000741The Colin de Verdière graph invariant.
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