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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000971
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 2
{{1},{2}}
=> 1
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 1
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> 2
{{1,3},{2,4}}
=> 3
{{1,3},{2},{4}}
=> 2
{{1,4},{2,3}}
=> 3
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 1
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 3
{{1,2,4,5},{3}}
=> 3
{{1,2,4},{3,5}}
=> 4
{{1,2,4},{3},{5}}
=> 3
{{1,2,5},{3,4}}
=> 4
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 3
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 2
{{1,3,4,5},{2}}
=> 2
{{1,3,4},{2,5}}
=> 4
{{1,3,4},{2},{5}}
=> 2
{{1,3,5},{2,4}}
=> 4
{{1,3},{2,4,5}}
=> 3
{{1,3},{2,4},{5}}
=> 3
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 2
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 4
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St001784
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001784: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 2
{{1},{2}}
=> 1
{{1,2,3}}
=> 2
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 3
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 1
{{1,2,3,4}}
=> 2
{{1,2,3},{4}}
=> 2
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> 3
{{1,3},{2,4}}
=> 3
{{1,3},{2},{4}}
=> 3
{{1,4},{2,3}}
=> 3
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 4
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 1
{{1,2,3,4,5}}
=> 2
{{1,2,3,4},{5}}
=> 2
{{1,2,3,5},{4}}
=> 2
{{1,2,3},{4,5}}
=> 2
{{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 2
{{1,3,4,5},{2}}
=> 3
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 3
{{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> 3
{{1,3},{2,4},{5}}
=> 3
{{1,3,5},{2},{4}}
=> 3
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> 3
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 4
Description
The minimum of the smallest closer and the second element of the block containing 1 in a set partition.
A closer of a set partition is the maximal element of a non-singleton block. This statistic is defined as $1$ if $\{1\}$ is a singleton block, and otherwise the minimum of the smallest closer and the second element of the block containing $1$.
Matching statistic: St000054
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => 2
{{1},{2}}
=> [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => 4
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: St000007
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 2
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [3,2,1] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => [3,1,2] => 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => [2,3,4,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => [4,2,1,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [2,4,3,1] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,4,1,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => [3,2,4,1] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => 3
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => [5,2,3,1,4] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => [2,3,5,4,1] => 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => [5,4,1,3,2] => 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => [4,5,1,2,3] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => [5,2,4,3,1] => 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => [3,2,4,5,1] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => [2,4,5,3,1] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => [5,2,1,3,4] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => [2,5,3,4,1] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => [3,4,2,5,1] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [5,4,3,1,2] => 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => [5,1,3,4,2] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,1,3,2,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => [4,3,1,5,2] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [4,3,2,5,1] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => [2,5,3,1,4] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [3,5,1,4,2] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => [3,5,2,4,1] => 3
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
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: St000193
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [[1]]
=> 1
{{1,2}}
=> [2,1] => [2,1] => [[0,1],[1,0]]
=> 2
{{1},{2}}
=> [1,2] => [1,2] => [[1,0],[0,1]]
=> 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> 4
Description
The row of the unique '1' in the first column of the alternating sign matrix.
Matching statistic: St000740
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => [1,2] => 2
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [2,1,3] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [3,1,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3,2,1] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [3,2,1,5,4] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [4,2,1,5,3] => 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [3,5,2,1,4] => 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [5,2,1,4,3] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [2,1,5,3,4] => 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [4,3,5,1,2] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [2,1,5,4,3] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [3,5,4,1,2] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [4,3,1,5,2] => 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [2,5,3,1,4] => 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [5,3,1,4,2] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [3,1,5,2,4] => 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [4,2,5,1,3] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [5,2,4,1,3] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [3,1,5,4,2] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [2,5,4,1,3] => 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [4,5,1,3,2] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [3,2,5,1,4] => 4
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001050
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> 1
{{1,2}}
=> [2,1] => [1,2] => {{1},{2}}
=> 2
{{1},{2}}
=> [1,2] => [2,1] => {{1,2}}
=> 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => {{1},{2,3}}
=> 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => {{1,2},{3}}
=> 2
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => {{1,3},{2}}
=> 2
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => {{1,2,3}}
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => {{1,2},{3,4}}
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 4
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000051
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Values
{{1}}
=> [1] => [1] => [.,.]
=> 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
{{1},{2}}
=> [1,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [[.,.],[.,.]]
=> 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> 3 = 4 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> 1 = 2 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> 3 = 4 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> 2 = 3 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
Description
The size of the left subtree of a binary tree.
Matching statistic: St000133
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,3,2] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [2,4,3,1] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 3 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => [4,1,3,2] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,5,4,3,1] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => [5,2,4,3,1] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => [2,5,4,1,3] => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,5,1,4,3] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => [4,2,5,3,1] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => [4,2,5,1,3] => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => [5,4,2,3,1] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => [2,5,4,1,3] => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => [2,5,4,3,1] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => [5,2,4,1,3] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,5,1,4,3] => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 3 = 4 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => [3,2,5,4,1] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => 2 = 3 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [3,2,5,1,4] => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [3,5,4,2,1] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => [3,1,5,4,2] => 3 = 4 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,5,4,2] => 3 = 4 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => [3,5,2,4,1] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => 2 = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => [3,2,5,1,4] => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => 2 = 3 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => 2 = 3 - 1
Description
The "bounce" of a permutation.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St000297The number of leading ones in a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001118The acyclic chromatic index of a graph. St000454The largest eigenvalue of a graph if it is integral. 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. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001060The distinguishing index of a graph. St000942The number of critical left to right maxima of the parking functions. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. 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. 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. St000668The least common multiple of the parts of the partition. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001330The hat guessing number of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module.
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