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Your data matches 77 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1
[[2,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 2
[[1,3]]
=> [1,2] => 1
[[2,3]]
=> [1,2] => 1
[[3,3]]
=> [1,2] => 1
[[1],[3]]
=> [2,1] => 2
[[2],[3]]
=> [2,1] => 2
[[1,1,2]]
=> [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => 3
[[1,2],[2]]
=> [2,1,3] => 2
[[1,4]]
=> [1,2] => 1
[[2,4]]
=> [1,2] => 1
[[3,4]]
=> [1,2] => 1
[[4,4]]
=> [1,2] => 1
[[1],[4]]
=> [2,1] => 2
[[2],[4]]
=> [2,1] => 2
[[3],[4]]
=> [2,1] => 2
[[1,1,3]]
=> [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => 3
[[1,2],[3]]
=> [3,1,2] => 3
[[1,3],[2]]
=> [2,1,3] => 2
[[1,3],[3]]
=> [2,1,3] => 2
[[2,2],[3]]
=> [3,1,2] => 3
[[2,3],[3]]
=> [2,1,3] => 2
[[1],[2],[3]]
=> [3,2,1] => 3
[[1,1,1,2]]
=> [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => 4
[[1,1,2],[2]]
=> [3,1,2,4] => 3
[[1,2,2],[2]]
=> [2,1,3,4] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => 3
[[1,5]]
=> [1,2] => 1
[[2,5]]
=> [1,2] => 1
[[3,5]]
=> [1,2] => 1
[[4,5]]
=> [1,2] => 1
[[5,5]]
=> [1,2] => 1
[[1],[5]]
=> [2,1] => 2
[[2],[5]]
=> [2,1] => 2
[[3],[5]]
=> [2,1] => 2
[[4],[5]]
=> [2,1] => 2
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000025
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: 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
St000025: 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
[[2,2]]
=> [1,2] => [1,0,1,0]
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 2
[[1,3]]
=> [1,2] => [1,0,1,0]
=> 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> 2
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> 2
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,4]]
=> [1,2] => [1,0,1,0]
=> 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> 2
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> 2
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> 2
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [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
[[1,2,2,2]]
=> [1,2,3,4] => [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
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[[1,5]]
=> [1,2] => [1,0,1,0]
=> 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> 2
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> 2
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> 2
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> 2
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000738
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [[1,2]]
=> 1
[[2,2]]
=> [1,2] => [[1,2]]
=> 1
[[1],[2]]
=> [2,1] => [[1],[2]]
=> 2
[[1,3]]
=> [1,2] => [[1,2]]
=> 1
[[2,3]]
=> [1,2] => [[1,2]]
=> 1
[[3,3]]
=> [1,2] => [[1,2]]
=> 1
[[1],[3]]
=> [2,1] => [[1],[2]]
=> 2
[[2],[3]]
=> [2,1] => [[1],[2]]
=> 2
[[1,1,2]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[1,2,2]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[2,2,2]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[1,1],[2]]
=> [3,1,2] => [[1,2],[3]]
=> 3
[[1,2],[2]]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[1,4]]
=> [1,2] => [[1,2]]
=> 1
[[2,4]]
=> [1,2] => [[1,2]]
=> 1
[[3,4]]
=> [1,2] => [[1,2]]
=> 1
[[4,4]]
=> [1,2] => [[1,2]]
=> 1
[[1],[4]]
=> [2,1] => [[1],[2]]
=> 2
[[2],[4]]
=> [2,1] => [[1],[2]]
=> 2
[[3],[4]]
=> [2,1] => [[1],[2]]
=> 2
[[1,1,3]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[1,2,3]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[1,3,3]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[2,2,3]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[2,3,3]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[3,3,3]]
=> [1,2,3] => [[1,2,3]]
=> 1
[[1,1],[3]]
=> [3,1,2] => [[1,2],[3]]
=> 3
[[1,2],[3]]
=> [3,1,2] => [[1,2],[3]]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[1,3],[3]]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[2,2],[3]]
=> [3,1,2] => [[1,2],[3]]
=> 3
[[2,3],[3]]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[[1,5]]
=> [1,2] => [[1,2]]
=> 1
[[2,5]]
=> [1,2] => [[1,2]]
=> 1
[[3,5]]
=> [1,2] => [[1,2]]
=> 1
[[4,5]]
=> [1,2] => [[1,2]]
=> 1
[[5,5]]
=> [1,2] => [[1,2]]
=> 1
[[1],[5]]
=> [2,1] => [[1],[2]]
=> 2
[[2],[5]]
=> [2,1] => [[1],[2]]
=> 2
[[3],[5]]
=> [2,1] => [[1],[2]]
=> 2
[[4],[5]]
=> [2,1] => [[1],[2]]
=> 2
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000740
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => 1
[[2,2]]
=> [1,2] => [2,1] => 1
[[1],[2]]
=> [2,1] => [1,2] => 2
[[1,3]]
=> [1,2] => [2,1] => 1
[[2,3]]
=> [1,2] => [2,1] => 1
[[3,3]]
=> [1,2] => [2,1] => 1
[[1],[3]]
=> [2,1] => [1,2] => 2
[[2],[3]]
=> [2,1] => [1,2] => 2
[[1,1,2]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,2]]
=> [1,2,3] => [3,2,1] => 1
[[2,2,2]]
=> [1,2,3] => [3,2,1] => 1
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => 3
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => 2
[[1,4]]
=> [1,2] => [2,1] => 1
[[2,4]]
=> [1,2] => [2,1] => 1
[[3,4]]
=> [1,2] => [2,1] => 1
[[4,4]]
=> [1,2] => [2,1] => 1
[[1],[4]]
=> [2,1] => [1,2] => 2
[[2],[4]]
=> [2,1] => [1,2] => 2
[[3],[4]]
=> [2,1] => [1,2] => 2
[[1,1,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[2,2,3]]
=> [1,2,3] => [3,2,1] => 1
[[2,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[3,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => 3
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => 3
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => 2
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => 2
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => 3
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => 2
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => 4
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => 3
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => 3
[[1,5]]
=> [1,2] => [2,1] => 1
[[2,5]]
=> [1,2] => [2,1] => 1
[[3,5]]
=> [1,2] => [2,1] => 1
[[4,5]]
=> [1,2] => [2,1] => 1
[[5,5]]
=> [1,2] => [2,1] => 1
[[1],[5]]
=> [2,1] => [1,2] => 2
[[2],[5]]
=> [2,1] => [1,2] => 2
[[3],[5]]
=> [2,1] => [1,2] => 2
[[4],[5]]
=> [2,1] => [1,2] => 2
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000051
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [[.,.],.]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [[.,.],.]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 3 - 1
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[[1,4]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[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] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 3 - 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 3 - 1
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[[1,5]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[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 size of the left subtree of a binary tree.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: 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
St000439: 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]
=> 2 = 1 + 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(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
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: 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
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: 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]
=> 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000026
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: 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
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[2,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [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
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [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
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
Description
The position of the first return of a Dyck path.
Matching statistic: St000061
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000061: Binary 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
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000061: Binary 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
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 3
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 3
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 3
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 3
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [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
[[1,2,2,2]]
=> [1,2,3,4] => [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
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 3
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [.,[.,.]]
=> 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [[.,.],.]
=> 2
Description
The number of nodes on the left branch of a binary tree.
Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path.
Matching statistic: St000066
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ 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
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ 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],[0,1]]
=> 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 3
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 2
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
The following 67 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000297The number of leading ones in a binary word. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000734The last entry in the first row of a standard tableau. St000839The largest opener of a set partition. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001497The position of the largest weak excedence of a permutation. St000141The maximum drop size of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000454The largest eigenvalue of a graph if it is integral. St001896The number of right descents of a signed permutations. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001946The number of descents in a parking function. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001568The smallest positive integer that does not appear twice in the partition. St001118The acyclic chromatic index of a graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. 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. 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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