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Your data matches 147 different statistics following compositions of up to 3 maps.
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Matching statistic: St000030
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000030: 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] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,3,4,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [4,3,1,2] => 3
[2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,3,4,1] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 4
[3,4,1,2] => [4,3,2,1] => 3
[3,4,2,1] => [3,4,2,1] => 3
[4,1,2,3] => [2,3,4,1] => 3
[4,1,3,2] => [2,4,3,1] => 3
[4,2,1,3] => [3,2,4,1] => 4
[4,2,3,1] => [3,4,2,1] => 3
[4,3,1,2] => [3,4,2,1] => 3
[4,3,2,1] => [4,3,2,1] => 3
Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000246
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 2
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 3
[1,2,4,3] => [1,4,3,2] => 3
[1,3,2,4] => [1,4,3,2] => 3
[1,3,4,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,4,3] => 4
[2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [2,4,1,3] => 3
[2,3,4,1] => [2,4,3,1] => 2
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,4,2] => 3
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 0
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000394
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,2] => [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001861
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001861: Signed 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] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => [3,4,1,2] => 4
[3,4,2,1] => [3,4,2,1] => 3
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [4,2,3,1] => 4
[4,3,1,2] => [4,3,1,2] => 3
[4,3,2,1] => [4,3,2,1] => 3
Description
The number of Bruhat lower covers of a permutation.
This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.
Matching statistic: St000018
(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
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000018: 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] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 2
[1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 3
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 3
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 3
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 3
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 3
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 4
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 4
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 3
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 3
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 2
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [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 $\pi$.
Matching statistic: St000029
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000029: 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}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 2
[3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 3
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 3
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 4
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => 3
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 4
Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000081
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,1] => ([],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[4,3,2,1] => ([],4)
=> ([],4)
=> 0
Description
The number of edges of a graph.
Matching statistic: St000136
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000136: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00305: Permutations —parking function⟶ Parking functions
St000136: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => [1,3,2] => 2
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 3
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 4
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 3
[2,3,4,1] => [2,4,3,1] => [2,4,3,1] => 2
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 3
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
Description
The dinv of a parking function.
Matching statistic: St000194
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000194: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00305: Permutations —parking function⟶ Parking functions
St000194: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => [1,3,2] => 2
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 3
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 3
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 4
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 3
[2,3,4,1] => [2,4,3,1] => [2,4,3,1] => 2
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 3
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
Description
The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function.
Matching statistic: St000224
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000224: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000224: 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}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 2
[3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 3
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 3
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 4
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => 3
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 4
Description
The sorting index of a permutation.
The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols.
Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.
The following 137 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001869The maximum cut size of a graph. St000004The major index of a permutation. St000067The inversion number of the alternating sign matrix. St000095The number of triangles of a graph. St000332The positive inversions of an alternating sign matrix. St000446The disorder of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001311The cyclomatic number of a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001727The number of invisible inversions of a permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001866The nesting alignments of a signed permutation. St001894The depth of a signed permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000957The number of Bruhat lower covers of a permutation. St000327The number of cover relations in a poset. St000728The dimension of a set partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000494The number of inversions of distance at most 3 of a permutation. St000809The reduced reflection length of the permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000794The mak of a permutation. St000795The mad of a permutation. St000797The stat`` of a permutation. St000833The comajor index of a permutation. St001077The prefix exchange distance of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000681The Grundy value of Chomp on Ferrers diagrams. St001596The number of two-by-two squares inside a skew partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St000993The multiplicity of the largest part of an integer partition. 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 $eA$. St001733The number of weak left to right maxima of a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000762The sum of the positions of the weak records of an integer composition. St001568The smallest positive integer that does not appear twice in the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000141The maximum drop size of a permutation. St000662The staircase size of the code of a permutation. St001626The number of maximal proper sublattices of a lattice. St000264The girth of a graph, which is not a tree. 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. St000260The radius of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000091The descent variation of a composition. St000173The segment statistic of a semistandard tableau. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001778The largest greatest common divisor of an element and its image in a permutation. St000422The energy of a graph, if it is integral. St001926Sparre Andersen's position of the maximum of a signed permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000567The sum of the products of all pairs of parts. St000706The product of the factorials of the multiplicities of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. 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 $eA$. 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000477The weight of a partition according to Alladi. St000997The even-odd crank of an integer partition. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra.
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