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Your data matches 87 different statistics following compositions of up to 3 maps.
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Matching statistic: St000372
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000372: 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] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[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] => 1
[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] => 1
[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
[5,1,2,3,4] => 2
[5,1,2,4,3] => 1
[5,1,3,2,4] => 2
[5,1,3,4,2] => 1
[5,1,4,2,3] => 1
[5,1,4,3,2] => 0
[5,2,1,3,4] => 1
[5,2,1,4,3] => 0
[5,2,3,1,4] => 1
[5,2,3,4,1] => 1
[5,2,4,1,3] => 0
[5,2,4,3,1] => 0
[5,3,1,2,4] => 1
[5,3,1,4,2] => 0
[5,3,2,1,4] => 0
[5,3,2,4,1] => 0
[5,3,4,1,2] => 0
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: St000223
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 1
[1,3,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 2
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => 1
[2,4,1,3] => [3,1,4,2] => 0
[2,4,3,1] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 0
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => 2
[5,1,2,4,3] => [1,5,4,2,3] => 1
[5,1,3,2,4] => [1,5,3,4,2] => 2
[5,1,3,4,2] => [1,5,3,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => 1
[5,1,4,3,2] => [1,5,2,3,4] => 0
[5,2,1,3,4] => [1,4,5,3,2] => 1
[5,2,1,4,3] => [1,4,5,2,3] => 0
[5,2,3,1,4] => [1,4,3,5,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => 0
[5,2,4,3,1] => [1,4,2,3,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => 1
[5,3,1,4,2] => [1,3,5,2,4] => 0
[5,3,2,1,4] => [1,3,4,5,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => 0
Description
The number of nestings in the permutation.
Matching statistic: St000371
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 1
[1,3,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 2
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => 1
[2,4,1,3] => [3,1,4,2] => 0
[2,4,3,1] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 0
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => 2
[5,1,2,4,3] => [1,5,4,2,3] => 1
[5,1,3,2,4] => [1,5,3,4,2] => 2
[5,1,3,4,2] => [1,5,3,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => 1
[5,1,4,3,2] => [1,5,2,3,4] => 0
[5,2,1,3,4] => [1,4,5,3,2] => 1
[5,2,1,4,3] => [1,4,5,2,3] => 0
[5,2,3,1,4] => [1,4,3,5,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => 0
[5,2,4,3,1] => [1,4,2,3,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => 1
[5,3,1,4,2] => [1,3,5,2,4] => 0
[5,3,2,1,4] => [1,3,4,5,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => 0
Description
The number of mid points of decreasing 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 $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000039
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ 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
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 1
[1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [3,2,1] => 0
[2,3,1] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [3,4,1,2] => 2
[1,2,4,3] => [4,3,1,2] => [3,4,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => 2
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 1
[1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => 0
[2,3,1,4] => [3,2,4,1] => [2,4,3,1] => 1
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 1
[2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 0
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 0
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,5,2,3] => 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,4,5,3,2] => 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,3,4,5,2] => 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,4,2,5,3] => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,4,2,3] => 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[5,2,3,1,4] => [1,4,3,5,2] => [1,3,5,4,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,2,4,3] => 0
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => [1,4,3,5,2] => 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 0
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,3,4,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => 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: St001866
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed 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
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 1
[1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [2,3,1] => 0
[2,3,1] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,2,1,4] => [2,3,4,1] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,2,1,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[5,2,3,1,4] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[5,3,2,1,4] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
* $-i < -j < -\pi(j) < -\pi(i)$, or
* $-i < j \leq \pi(j) < -\pi(i)$, or
* $i < j \leq \pi(j) < \pi(i)$.
Matching statistic: St000356
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000356: 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] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [1,3,2] => 1
[1,3,2] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[2,1,3] => [2,3,1] => [3,2,1] => [1,2,3] => 0
[2,3,1] => [2,1,3] => [2,1,3] => [3,1,2] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 2
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 2
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 0
[2,1,3,4] => [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 0
[2,3,1,4] => [3,2,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 1
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [2,1,3,4] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 0
[3,1,2,4] => [2,4,3,1] => [3,4,2,1] => [1,2,4,3] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,1,2,4] => 0
[3,2,1,4] => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => [2,5,3,4,1] => 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,3,4,5,2] => [2,5,4,3,1] => 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,4,5,2,3] => [3,2,5,4,1] => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => 0
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,3,4,2] => [2,4,3,5,1] => 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,2,4,3] => [3,4,2,5,1] => 0
[5,2,3,1,4] => [1,4,3,5,2] => [1,3,5,4,2] => [2,4,5,3,1] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,4,2,3] => [3,2,4,5,1] => 0
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0
[5,3,1,2,4] => [1,3,5,4,2] => [1,4,5,3,2] => [2,3,5,4,1] => 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => 0
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[5,3,2,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 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: St000358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000358: 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
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[1,3,2] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[2,1,3] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [3,2,4,1] => [4,1,3,2] => 2
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => [4,2,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 2
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => [3,4,1,2] => 1
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => [2,4,1,3] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[2,1,3,4] => [3,4,2,1] => [4,2,3,1] => [3,1,4,2] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [3,2,4,1] => 0
[2,3,1,4] => [3,2,4,1] => [2,4,3,1] => [4,3,1,2] => 1
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [2,4,3,1] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,4,2,1] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,4,2,1] => 0
[3,2,1,4] => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,5,2,4,3] => 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,4,2,5,3] => [1,5,3,4,2] => 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => [1,4,5,2,3] => 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,4,5,2,3] => [1,3,5,2,4] => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,3,4,2] => [1,4,2,5,3] => 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,4,3,5,2] => 0
[5,2,3,1,4] => [1,4,3,5,2] => [1,3,5,4,2] => [1,5,4,2,3] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,4,2,3] => [1,3,5,4,2] => 0
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => [1,4,5,3,2] => [1,5,3,2,4] => 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,4,5,3,2] => 0
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001687
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001687: 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] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [3,2,1] => [2,1,3] => [2,1,3] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [3,2,1] => [2,3,1] => 0
[3,1,2] => [2,1,3] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 2
[1,2,4,3] => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,1,4] => [3,2,1,4] => 2
[1,3,4,2] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,4,2,3] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [4,3,1,2] => [4,2,1,3] => [2,4,1,3] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[2,3,1,4] => [4,1,3,2] => [4,3,1,2] => [3,1,4,2] => 1
[2,3,4,1] => [1,4,3,2] => [4,3,2,1] => [3,2,4,1] => 1
[2,4,1,3] => [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 0
[2,4,3,1] => [1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 0
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,2,1,4] => [4,1,2,3] => [3,4,1,2] => [4,1,3,2] => 0
[3,2,4,1] => [1,4,2,3] => [3,4,2,1] => [4,2,3,1] => 0
[3,4,1,2] => [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[3,4,2,1] => [1,2,4,3] => [2,4,3,1] => [3,4,2,1] => 0
[4,1,2,3] => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 1
[4,1,3,2] => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[4,2,1,3] => [3,1,2,4] => [3,1,4,2] => [4,3,1,2] => 0
[4,2,3,1] => [1,3,2,4] => [3,2,4,1] => [2,4,3,1] => 0
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[5,1,2,4,3] => [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => 1
[5,1,3,2,4] => [4,2,3,1,5] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[5,1,3,4,2] => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[5,1,4,2,3] => [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[5,1,4,3,2] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,1,3,4] => [4,3,1,2,5] => [4,2,1,5,3] => [2,5,4,1,3] => 1
[5,2,1,4,3] => [3,4,1,2,5] => [4,1,2,5,3] => [5,4,1,2,3] => 0
[5,2,3,1,4] => [4,1,3,2,5] => [4,3,1,5,2] => [3,1,5,4,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [4,3,2,5,1] => [3,2,5,4,1] => 1
[5,2,4,1,3] => [3,1,4,2,5] => [4,1,3,5,2] => [3,5,4,1,2] => 0
[5,2,4,3,1] => [1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,4,1] => 0
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,5,3] => [5,4,2,1,3] => 1
[5,3,1,4,2] => [2,4,1,3,5] => [1,4,2,5,3] => [1,5,4,2,3] => 0
[5,3,2,1,4] => [4,1,2,3,5] => [3,4,1,5,2] => [5,4,1,3,2] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [3,4,2,5,1] => [5,4,2,3,1] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [1,4,3,5,2] => [1,3,5,4,2] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St001862
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed 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
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [2,3,1] => 1
[1,3,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
[1,2,4,3] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 2
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[1,4,2,3] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,3,1,4] => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[2,4,1,3] => [3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,2,3,1,4] => [1,4,3,5,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,1,2,4] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
* $i < j \leq \pi(i) < \pi(j)$, or
* $-i < j \leq -\pi(i) < \pi(j)$, or
* $i > j > \pi(i) > \pi(j)$.
Matching statistic: St001845
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
[3,2,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 2
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[5,1,2,3,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 2
[5,1,2,4,3] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[5,1,3,2,4] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
[5,1,3,4,2] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[5,1,4,2,3] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[5,1,4,3,2] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[5,2,1,3,4] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[5,2,1,4,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
[5,2,3,1,4] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[5,2,4,3,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[5,3,1,2,4] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[5,3,1,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[5,3,2,1,4] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[5,3,2,4,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[5,3,4,1,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[5,3,4,2,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[5,4,1,2,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[5,4,1,3,2] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[5,4,2,1,3] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[5,4,2,3,1] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[5,4,3,1,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
The following 77 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. 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. St001128The exponens consonantiae of a partition. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. 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. 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. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. 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. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001857The number of edges in the reduced word graph of a signed permutation. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000365The number of double ascents of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001130The number of two successive successions in a permutation. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. 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 $eA$.
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