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Your data matches 122 different statistics following compositions of up to 3 maps.
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Matching statistic: St001549
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 0
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St000039
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 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: St000232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => {{1}}
=> 0
[.,[.,.]]
=> [2,1] => {{1,2}}
=> 0
[[.,.],.]
=> [1,2] => {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> 0
[.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> 0
[[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> 0
[[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> 0
[[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000358
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ 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
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,4,2,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,2,5,3,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,3,5,2,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,3,4,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,4,5,3,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,5,3,2,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,4,2,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,5,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,3,4,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,5,3,2] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,5,4,2] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 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: St000123
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,4,1,2] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [3,1,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,5,1,2] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [3,4,5,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [4,5,2,1,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [4,5,1,3,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,5,2,3,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,1,5,2,3] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [4,1,5,3,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,1,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,1,2,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,5,1,2,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [3,5,2,1,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,1,5,2,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,1,3,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,5,2,3] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,4,2,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,2,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
* The Simion-Schmidt map takes a permutation and turns each occurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
* It is the number of pairs of positions for the pattern letters 2 and 1 in occurrences of 321 in a permutation. Thus, for a permutation $\pi$ this is the number of pairs $(j,k)$ such that there exists an index $i$ satisfying $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. See also [[St000119]] and [[St000371]].
* Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000223
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,4,1,2] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [3,1,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,5,1,2] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [3,4,5,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [4,5,2,1,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [4,5,1,3,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,5,2,3,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,1,5,2,3] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [4,1,5,3,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,1,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,1,2,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,5,1,2,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [3,5,2,1,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,1,5,2,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,1,3,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,5,2,3] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,4,2,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,2,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
Description
The number of nestings in the permutation.
Matching statistic: St000233
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => {{1}}
=> {{1}}
=> 0
[.,[.,.]]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0
[[.,.],.]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> 0
[.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[[.,.],[.,.]]
=> [1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> 0
[[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => {{1},{2,4},{3,5}}
=> {{1},{2,5},{3,4}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 0
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000356
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ 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
Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
=> [1,2] => [1,2] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => [2,1,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [1,2,3] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,4,2,1] => [2,1,3,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => [3,2,4,1,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => [3,1,4,2,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,2,5,3,1] => [2,4,1,3,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => [1,4,2,3,5] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,3,5,2,1] => [2,3,1,4,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,4,5,3,1] => [4,2,1,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,5,3,2,1] => [2,1,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => [3,4,1,2,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,5,3,2] => [5,2,1,3,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => [4,3,5,1,2] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 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: St001685
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
=> [1,2] => [1,2] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => [2,1,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [1,2,3] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,4,2,1] => [2,1,3,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => [3,2,4,1,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => [3,1,4,2,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,2,5,3,1] => [2,4,1,3,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [4,3,5,2,1] => [2,3,1,4,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,4,5,3,1] => [4,2,1,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,5,3,2,1] => [2,1,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => [3,4,1,2,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => [1,2,4,3,5] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,5,3,2] => [5,2,1,3,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => [4,3,5,1,2] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000516
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => ? = 0
[.,[.,.]]
=> [2,1] => [1,2] => [2,1] => 0
[[.,.],.]
=> [1,2] => [2,1] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [2,1,3] => 0
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [3,2,1] => 0
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [2,3,1,4] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [2,1,4,3] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [2,1,3,4] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [3,2,4,1] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [3,2,1,4] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [4,2,3,1] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [4,3,2,1] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,3,4,2] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [1,4,3,2] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [2,3,5,4,1] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [2,3,1,5,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [2,3,1,4,5] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [2,4,3,5,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [2,4,3,1,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [2,5,3,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [2,5,4,3,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [2,1,4,5,3] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [2,1,4,3,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [2,1,5,4,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [2,1,3,5,4] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [2,1,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [3,2,4,5,1] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [3,2,5,4,1] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [3,2,1,5,4] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [3,2,1,4,5] => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [4,2,3,5,1] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [4,2,3,1,5] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [4,3,2,5,1] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [4,3,2,1,5] => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [5,2,3,4,1] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [5,2,4,3,1] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [5,3,2,4,1] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [5,4,2,3,1] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [5,4,3,2,1] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,3,4,5,2] => 0
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
The following 112 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000710The number of big deficiencies of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001586The number of odd parts smaller than the largest even part in an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001964The interval resolution global dimension of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St000929The constant term of the character polynomial of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000940The number of characters of the symmetric group whose value on the partition is zero. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St000731The number of double exceedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000031The number of cycles in the cycle decomposition of a permutation. St000655The length of the minimal rise of a Dyck path. St001513The number of nested exceedences of a permutation. St000302The determinant of the distance matrix of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. 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. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. 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. 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. St001845The number of join irreducibles minus the rank of a lattice. St000741The Colin de Verdière graph invariant. St000068The number of minimal elements in a poset. St001490The number of connected components of a skew partition. St000842The breadth of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St000629The defect of a binary word. St001875The number of simple modules with projective dimension at most 1. St000383The last part of an integer composition. St001867The number of alignments of type EN of a signed permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001570The minimal number of edges to add to make a graph Hamiltonian. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. 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 $K[x]/(x^n)$. St001344The neighbouring number of a permutation. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001584The area statistic between a Dyck path and its bounce path. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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