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Your data matches 74 different statistics following compositions of up to 3 maps.
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Matching statistic: St000141
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(load all 9 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 0
[[2,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 1
[[1,3]]
=> [1,2] => 0
[[2,3]]
=> [1,2] => 0
[[3,3]]
=> [1,2] => 0
[[1],[3]]
=> [2,1] => 1
[[2],[3]]
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => 2
[[1,2],[2]]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => 0
[[2,4]]
=> [1,2] => 0
[[3,4]]
=> [1,2] => 0
[[4,4]]
=> [1,2] => 0
[[1],[4]]
=> [2,1] => 1
[[2],[4]]
=> [2,1] => 1
[[3],[4]]
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => 2
[[1,2],[3]]
=> [3,1,2] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => 2
[[2,3],[3]]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2
[[1,5]]
=> [1,2] => 0
[[2,5]]
=> [1,2] => 0
[[3,5]]
=> [1,2] => 0
[[4,5]]
=> [1,2] => 0
[[5,5]]
=> [1,2] => 0
[[1],[5]]
=> [2,1] => 1
[[2],[5]]
=> [2,1] => 1
[[3],[5]]
=> [2,1] => 1
[[4],[5]]
=> [2,1] => 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000442
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(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ 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
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000730
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(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => {{1},{2}}
=> 0
[[2,2]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[2]]
=> [2,1] => {{1,2}}
=> 1
[[1,3]]
=> [1,2] => {{1},{2}}
=> 0
[[2,3]]
=> [1,2] => {{1},{2}}
=> 0
[[3,3]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[3]]
=> [2,1] => {{1,2}}
=> 1
[[2],[3]]
=> [2,1] => {{1,2}}
=> 1
[[1,1,2]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,2,2]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[2,2,2]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,1],[2]]
=> [3,1,2] => {{1,3},{2}}
=> 2
[[1,2],[2]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[1,4]]
=> [1,2] => {{1},{2}}
=> 0
[[2,4]]
=> [1,2] => {{1},{2}}
=> 0
[[3,4]]
=> [1,2] => {{1},{2}}
=> 0
[[4,4]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[4]]
=> [2,1] => {{1,2}}
=> 1
[[2],[4]]
=> [2,1] => {{1,2}}
=> 1
[[3],[4]]
=> [2,1] => {{1,2}}
=> 1
[[1,1,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[2,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[2,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[3,3,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[[1,1],[3]]
=> [3,1,2] => {{1,3},{2}}
=> 2
[[1,2],[3]]
=> [3,1,2] => {{1,3},{2}}
=> 2
[[1,3],[2]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[1,3],[3]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[2,2],[3]]
=> [3,1,2] => {{1,3},{2}}
=> 2
[[2,3],[3]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[1],[2],[3]]
=> [3,2,1] => {{1,3},{2}}
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[[1,5]]
=> [1,2] => {{1},{2}}
=> 0
[[2,5]]
=> [1,2] => {{1},{2}}
=> 0
[[3,5]]
=> [1,2] => {{1},{2}}
=> 0
[[4,5]]
=> [1,2] => {{1},{2}}
=> 0
[[5,5]]
=> [1,2] => {{1},{2}}
=> 0
[[1],[5]]
=> [2,1] => {{1,2}}
=> 1
[[2],[5]]
=> [2,1] => {{1,2}}
=> 1
[[3],[5]]
=> [2,1] => {{1,2}}
=> 1
[[4],[5]]
=> [2,1] => {{1,2}}
=> 1
Description
The maximal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
Matching statistic: St000013
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(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ 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
St000013: Dyck paths ⟶ 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 = 0 + 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000306
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ 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
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000662
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ 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
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001046
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ 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
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
Description
The maximal number of arcs nesting a given arc of a perfect matching.
This is also the largest weight of a down step in the histoire d'Hermite corresponding to the perfect matching.
Matching statistic: St001205
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001205: Dyck paths ⟶ 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
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
Description
The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Nakayama algebra and the relation to Dyck paths.
Matching statistic: St000062
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ 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
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3 = 2 + 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ 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
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ 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 = 0 + 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 1 + 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 3 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 3 = 2 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 3 = 2 + 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
Description
The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St001589The nesting number of a perfect matching. St000454The largest eigenvalue of a graph if it is integral. St001769The reflection length of a 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 $K[x]/(x^n)$. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000379The number of Hamiltonian cycles in a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001568The smallest positive integer that does not appear twice in the partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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