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Your data matches 103 different statistics following compositions of up to 3 maps.
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Matching statistic: St000748
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000748: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 3
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 2
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 3
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
Description
The major index of the permutation obtained by flattening the set partition.
A set partition can be represented by a sequence of blocks where the first entries of the blocks and the blocks themselves are increasing. This statistic is then the major index of the permutation obtained by flattening the set partition in this canonical form.
Matching statistic: St000004
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000463
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [2,3,1] => 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [4,2,1,3] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,4,2] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [3,1,4,2] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The number of admissible inversions of a permutation.
Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$.
An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions:
$1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.
Matching statistic: St000653
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000673
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St000794
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000794: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000794: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The mak of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(2\underline{31})$, $(\underline{32}1)$, $(1\underline{32})$, $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St001005
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
Matching statistic: St001696
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [[1],[2]]
=> 0
{{1},{2}}
=> [1,2] => [[1,2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [[1,2],[3]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [[1,3],[2]]
=> 2
{{1,3},{2}}
=> [3,2,1] => [[1],[2],[3]]
=> 0
{{1},{2,3}}
=> [1,3,2] => [[1,2],[3]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [[1,2,3]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [[1,2,3],[4]]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [[1,2,4],[3]]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [[1,2],[3,4]]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [[1,2,4],[3]]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
Description
The natural major index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000008
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [2,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [3,1] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [3,1] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [2,2] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [2,2] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [2,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [2,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [3,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4] => 0
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [2,4,1,3] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [2,3,1,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [2,1,4,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [2,1,4,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [2,3,4,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
The following 93 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000156The Denert index of a permutation. St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000305The inverse major index of a permutation. St000330The (standard) major index of a standard tableau. St000391The sum of the positions of the ones in a binary word. St000462The major index minus the number of excedences of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000693The modular (standard) major index of a standard tableau. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000833The comajor index of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000915The Ore degree of a graph. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001090The number of pop-stack-sorts needed to sort a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001279The sum of the parts of an integer partition that are at least two. St001375The pancake length of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001671Haglund's hag of a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000058The order of a permutation. St000240The number of indices that are not small excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000485The length of the longest cycle of a permutation. St000652The maximal difference between successive positions of a permutation. St000738The first entry in the last row of a standard tableau. St000763The sum of the positions of the strong records of an integer composition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000422The energy of a graph, if it is integral. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St000699The toughness times the least common multiple of 1,. St001058The breadth of the ordered tree. St000674The number of hills of a Dyck path. 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000455The second largest eigenvalue of a graph if it is integral. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000259The diameter of a connected graph. St001651The Frankl number of a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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