Your data matches 233 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000034
(load all 14 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000034: 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
{{1},{2}}
 => [[1],[2]]
 => [2,1] => 0
{{1,2,3}}
 => [[1,2,3]]
 => [1,2,3] => 0
{{1,2},{3}}
 => [[1,2],[3]]
 => [3,1,2] => 0
{{1,3},{2}}
 => [[1,3],[2]]
 => [2,1,3] => 0
{{1},{2,3}}
 => [[1,3],[2]]
 => [2,1,3] => 0
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [3,2,1] => 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [4,1,2,3] => 0
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [3,1,2,4] => 0
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => 0
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => 0
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000039
(load all 21 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => 1
{{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] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,4,1,3] => 1
{{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] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 1
{{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] => [3,4,1,2] => 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] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => 1
{{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 crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000356
(load all 22 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => 1
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,1,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,4,2,3] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,4,3,2] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [4,1,3,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4,1,2,3] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,3,1,4] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,2,4,3] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,3,2,4] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,3,4,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,1,3] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1,2,4] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,2,3,1] => 0
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000372
(load all 30 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 1
Description
The number of mid points of increasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$. The generating function is given by [1].
Matching statistic: St001083
(load all 25 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => 1
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,1,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,4,2,3] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,4,3,2] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [4,1,3,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4,1,2,3] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [2,3,1,4] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,2,4,3] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,3,2,4] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,3,4,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,1,3] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1,2,4] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,2,3,1] => 0
Description
The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001682
(load all 39 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001682: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.
Matching statistic: St001685
(load all 24 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => 1
{{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] => [1,2,4,3] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,3,2,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,2,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 1
{{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] => [2,4,1,3] => 1
{{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] => 1
{{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 distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St001727
(load all 20 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,4,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,4,2,3] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,2,1,4] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,4,3,1] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,2,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,4,3,2] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,1,2] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [2,3,1,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,4,3] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,2,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,4,2,1] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,4,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
Description
The number of invisible inversions of a permutation. A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St001931
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001931: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2] => 0
{{1},{2}}
 => [1,2] => [2] => 0
{{1,2,3}}
 => [2,3,1] => [3] => 0
{{1,2},{3}}
 => [2,1,3] => [3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,2] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,1] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4] => 0
Description
The weak major index of an integer composition regarded as a word. This is the sum of the positions of the weak descents, regarding the composition as a word. That is, for a composition $c = (c_1,\dots,c_n)$, $$ \sum_{\substack{1\leq i < n\\ c_i\geq c_{i+1}}} i. $$
Matching statistic: St000002
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [1,3,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [1,2,3] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,3,2] => [2,3,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,4,1,3] => [4,3,2,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => [1,4,3,2] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [4,1,3,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,4,3] => [4,1,3,2] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [2,3,1,4] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [1,2,4,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [1,3,2,4] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [3,1,2,4] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,4,3,2] => [3,1,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [3,1,4,2] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [3,1,2,4] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,4,3,2] => [3,1,2,4] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,4,3,2] => [3,1,2,4] => 1
Description
The number of occurrences of the pattern 123 in a permutation.
The following 223 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000057The Shynar inversion number of a standard tableau. St000091The descent variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000204The number of internal nodes of a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000223The number of nestings in the permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000446The disorder of a permutation. St000534The number of 2-rises of a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000663The number of right floats of a permutation. St000682The Grundy value of Welter's game on a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001565The number of arithmetic progressions of length 2 in a permutation. St001623The number of doubly irreducible elements of a lattice. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000816The number of standard composition tableaux of the composition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000293The number of inversions of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001811The Castelnuovo-Mumford regularity of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001220The width of a permutation. St000290The major index of a binary word. St000369The dinv deficit of a Dyck path. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001485The modular major index of a binary word. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000886The number of permutations with the same antidiagonal sums. St001487The number of inner corners of a skew partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001964The interval resolution global dimension of a poset. St000010The length of the partition. St000012The area of a Dyck path. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001127The sum of the squares of the parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001651The Frankl number of a lattice. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000454The largest eigenvalue of a graph if it is integral. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. 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. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001857The number of edges in the reduced word graph of a signed permutation. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of 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. St000225Difference between largest and smallest parts in a partition. St000455The second largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. 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. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000102The charge of a semistandard tableau. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.