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Your data matches 142 different statistics following compositions of up to 3 maps.
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Matching statistic: St001084
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St001084: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 2
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 3
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 3
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
Description
The number of occurrences of the vincular pattern |1-23 in a permutation.
This is the number of occurrences of the pattern $123$, where the first two matched entries are the first two entries of the permutation.
In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.
Matching statistic: St001745
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001745: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001745: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 0
[1,2,4,3] => [1,4,3,2] => 0
[1,3,2,4] => [1,4,3,2] => 0
[1,3,4,2] => [1,4,3,2] => 0
[1,4,2,3] => [1,4,3,2] => 0
[1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,4,1,3] => 0
[2,3,4,1] => [2,4,3,1] => 0
[2,4,1,3] => [2,4,1,3] => 0
[2,4,3,1] => [2,4,3,1] => 0
[3,1,2,4] => [3,1,4,2] => 1
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,3,2] => 0
[4,1,3,2] => [4,1,3,2] => 0
[4,2,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => 0
[1,2,3,5,4] => [1,5,4,3,2] => 0
[1,2,4,3,5] => [1,5,4,3,2] => 0
[1,2,4,5,3] => [1,5,4,3,2] => 0
[1,2,5,3,4] => [1,5,4,3,2] => 0
[1,2,5,4,3] => [1,5,4,3,2] => 0
[1,3,2,4,5] => [1,5,4,3,2] => 0
[1,3,2,5,4] => [1,5,4,3,2] => 0
[1,3,4,2,5] => [1,5,4,3,2] => 0
[1,3,4,5,2] => [1,5,4,3,2] => 0
[1,3,5,2,4] => [1,5,4,3,2] => 0
[1,3,5,4,2] => [1,5,4,3,2] => 0
[1,4,2,3,5] => [1,5,4,3,2] => 0
[1,4,2,5,3] => [1,5,4,3,2] => 0
[1,4,3,2,5] => [1,5,4,3,2] => 0
[1,4,3,5,2] => [1,5,4,3,2] => 0
[1,4,5,2,3] => [1,5,4,3,2] => 0
Description
The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000355
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,3,1] => [2,3,1] => 0
[2,3,1] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,3,2] => 0
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 0
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 0
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => [2,1,4,3] => 2
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 0
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 0
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000359
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,3,2] => 0
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 0
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 0
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 0
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 0
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 0
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => [2,1,4,3] => 0
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 0
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 0
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 0
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000473
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> []
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> []
=> 0
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 0
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000804
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000804: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000804: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => 0
[2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,1,3] => 0
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 0
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 0
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 0
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 0
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0
[2,3,1,4] => [2,4,1,3] => [1,4,2,3] => 1
[2,3,4,1] => [2,4,3,1] => [1,3,2,4] => 2
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [4,1,3,2] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => 0
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => 0
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => 0
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => 0
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,1,4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => 0
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
Description
The number of occurrences of the vincular pattern |123 in a permutation.
This is the number of occurrences of the pattern $(1,2,3)$, such that the letter matched by $1$ is the first entry of the permutation.
Matching statistic: St000711
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000711: 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
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> [2,1] => 0
[2,1] => [2,1] => [1,1,0,0]
=> [1,2] => 0
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 0
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000801
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => ? = 0
[1,2] => [1,0,1,0]
=> [2,1] => [2,1] => 0
[2,1] => [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,4,3,1] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,4,1,3] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,4,3] => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,4,3] => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,4,3,2] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,4,3,2] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 0
Description
The number of occurrences of the vincular pattern |312 in a permutation.
This is the number of occurrences of the pattern $(3,1,2)$, such that the letter matched by $3$ is the first entry of the permutation.
Matching statistic: St000802
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000802: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000802: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,3,2] => [2,3,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [2,3,1] => [1,3,2] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 0
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 1
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [4,2,3,1] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 1
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [4,2,3,1] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [1,4,2,3] => 0
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [1,4,2,3] => 0
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [3,4,1,2] => 0
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [3,4,2,1] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [1,4,3,2] => 0
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [2,4,3,1] => 0
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [1,2,4,3] => 0
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [1,2,4,3] => 0
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [3,2,4,1] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,3,4,2] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
Description
The number of occurrences of the vincular pattern |321 in a permutation.
This is the number of occurrences of the pattern $(3,2,1)$, such that the letter matched by $3$ is the first entry of the permutation.
Matching statistic: St000803
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => ? = 0
[1,2] => [1,0,1,0]
=> [2,1] => [1,2] => 0
[2,1] => [1,1,0,0]
=> [1,2] => [2,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [2,1,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1,2,4] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1,4] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,3,4] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,1,3] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,1,4,3] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,4,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,4,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,4,2] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,4,2] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,1,2,5] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,1,5] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,2,1,3,5] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1,2,3,5] => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1,2,3,5] => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,4,2,1,5] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,4,1,2,5] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1,5] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,4,5] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,2,4,5] => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,2,4,5] => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1,5] => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,4,5] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1,5] => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,4,5] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,4,5] => 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,4,5] => 0
Description
The number of occurrences of the vincular pattern |132 in a permutation.
This is the number of occurrences of the pattern $(1,3,2)$, such that the letter matched by $1$ is the first entry of the permutation.
The following 132 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001586The number of odd parts smaller than the largest even part in an integer partition. St000117The number of centered tunnels of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001175The size of a partition minus the hook length of the base cell. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001541The Gini index of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000929The constant term of the character polynomial of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001596The number of two-by-two squares inside a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000699The toughness times the least common multiple of 1,. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001964The interval resolution global dimension of a poset. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. St001571The Cartan determinant of the integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001557The number of inversions of the second entry of a permutation. 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. St000708The product of the parts of an integer partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001868The number of alignments of type NE of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001846The number of elements which do not have a complement in the lattice. St000527The width of the poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001867The number of alignments of type EN of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation.
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