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Your data matches 23 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: St000232
(load all 2 compositions to match this statistic)
(load all 2 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: 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: St000358
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●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
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [1,6,3,7,5,4,2] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [1,7,3,6,5,4,2] => ? = 3
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,3,7,6,5,4,2] => [1,5,6,4,7,3,2] => ? = 0
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [1,5,7,4,6,3,2] => ? = 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => [1,6,4,7,5,3,2] => ? = 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [1,7,5,4,6,3,2] => ? = 1
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [1,7,4,6,5,3,2] => ? = 2
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,3,4,7,6,5,2] => [1,6,5,7,4,3,2] => ? = 0
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [1,7,5,6,4,3,2] => ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => [1,6,7,5,3,4,2] => ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => [1,4,6,7,5,3,2] => ? = 0
[[.,.],[[[[.,.],.],.],[.,.]]]
=> [1,3,4,5,7,6,2] => [1,6,7,5,4,3,2] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [1,7,6,3,5,4,2] => ? = 2
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [1,3,6,5,4,7,2] => [1,5,4,7,6,3,2] => ? = 0
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [1,7,6,4,5,3,2] => ? = 1
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,3,4,6,5,7,2] => [1,5,7,6,4,3,2] => ? = 0
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [1,7,6,5,3,4,2] => ? = 1
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,3,5,4,6,7,2] => [1,4,7,6,5,3,2] => ? = 0
[[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => ? = 0
[[[.,.],[.,[[.,.],.]]],[.,.]]
=> [1,4,5,3,2,7,6] => [1,5,3,4,2,7,6] => ? = 1
[[[.,.],[[[.,.],.],.]],[.,.]]
=> [1,3,4,5,2,7,6] => [1,5,4,3,2,7,6] => ? = 0
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,4,6,5,3,2,7] => [1,5,3,6,4,2,7] => ? = 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,4,5,6,3,2,7] => [1,6,3,5,4,2,7] => ? = 2
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,3,6,5,4,2,7] => [1,5,4,6,3,2,7] => ? = 0
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,3,5,6,4,2,7] => [1,6,4,5,3,2,7] => ? = 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,3,4,6,5,2,7] => [1,5,6,4,3,2,7] => ? = 0
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,4,5,3,6,2,7] => [1,6,5,3,4,2,7] => ? = 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => [1,6,5,4,3,2,7] => ? = 0
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,4,5,3,2,6,7] => [1,5,3,4,2,6,7] => ? = 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => ? = 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: St000356
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●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
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [1,6,3,7,5,4,2] => [7,2,5,1,3,4,6] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [1,7,3,6,5,4,2] => [7,1,5,2,3,4,6] => ? = 3
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [1,5,7,4,6,3,2] => [7,3,1,4,2,5,6] => ? = 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => [1,6,4,7,5,3,2] => [7,2,4,1,3,5,6] => ? = 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [1,7,5,4,6,3,2] => [7,1,3,4,2,5,6] => ? = 1
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [1,7,4,6,5,3,2] => [7,1,4,2,3,5,6] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,3,6,5,7,4,2] => [7,5,2,3,1,4,6] => ? = 0
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => [1,3,7,5,6,4,2] => [7,5,1,3,2,4,6] => ? = 1
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [1,7,5,6,4,3,2] => [7,1,3,2,4,5,6] => ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => [1,6,7,5,3,4,2] => [7,2,1,3,5,4,6] => ? = 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => [1,3,6,7,5,4,2] => [7,5,2,1,3,4,6] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [1,7,6,3,5,4,2] => [7,1,2,5,3,4,6] => ? = 2
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [1,7,6,4,5,3,2] => [7,1,2,4,3,5,6] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => [1,3,5,7,6,4,2] => [7,5,3,1,2,4,6] => ? = 0
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [1,7,6,5,3,4,2] => [7,1,2,3,5,4,6] => ? = 1
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,7,6,5,4,3] => [1,2,5,6,4,7,3] => [7,6,3,2,4,1,5] => ? = 0
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [1,2,6,7,5,4,3] => [1,2,5,7,4,6,3] => [7,6,3,1,4,2,5] => ? = 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,2,5,7,6,4,3] => [1,2,6,4,7,5,3] => [7,6,2,4,1,3,5] => ? = 1
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,2,6,5,7,4,3] => [1,2,7,5,4,6,3] => [7,6,1,3,4,2,5] => ? = 1
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,2,5,6,7,4,3] => [1,2,7,4,6,5,3] => [7,6,1,4,2,3,5] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,2,4,7,6,5,3] => [1,2,6,5,7,4,3] => [7,6,2,3,1,4,5] => ? = 0
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,2,4,6,7,5,3] => [1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,2,5,4,7,6,3] => [1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 0
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [1,2,4,5,7,6,3] => [1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => ? = 0
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [1,2,6,5,4,7,3] => [1,2,5,4,7,6,3] => [7,6,3,4,1,2,5] => ? = 0
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [1,2,5,6,4,7,3] => [1,2,7,6,4,5,3] => [7,6,1,2,4,3,5] => ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,2,4,6,5,7,3] => [1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => ? = 0
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,2,5,4,6,7,3] => [1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 0
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,3,2,7,6,5,4] => [1,3,2,6,5,7,4] => [7,5,6,2,3,1,4] => ? = 0
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> [1,3,2,6,7,5,4] => [1,3,2,7,5,6,4] => [7,5,6,1,3,2,4] => ? = 1
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,3,2,5,7,6,4] => [1,3,2,6,7,5,4] => [7,5,6,2,1,3,4] => ? = 0
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [1,3,2,6,5,7,4] => [1,3,2,5,7,6,4] => [7,5,6,3,1,2,4] => ? = 0
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 0
[[[[.,.],.],.],[.,[[.,.],.]]]
=> [1,2,3,6,7,5,4] => [1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 1
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [1,2,3,5,7,6,4] => [1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 0
[[[[.,.],.],.],[[.,[.,.]],.]]
=> [1,2,3,6,5,7,4] => [1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 0
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,4,3,2,7,6,5] => [1,3,4,2,6,7,5] => [7,5,4,6,2,1,3] => ? = 0
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,4,3,2,6,7,5] => [1,3,4,2,7,6,5] => [7,5,4,6,1,2,3] => ? = 0
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> [1,3,4,2,7,6,5] => [1,4,3,2,6,7,5] => [7,4,5,6,2,1,3] => ? = 0
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 0
[[[[.,.],[.,.]],.],[.,[.,.]]]
=> [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => [7,5,6,4,2,1,3] => ? = 0
[[[[.,.],[.,.]],.],[[.,.],.]]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => [7,5,6,4,1,2,3] => ? = 0
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 0
[[[.,.],[.,[[.,.],.]]],[.,.]]
=> [1,4,5,3,2,7,6] => [1,5,3,4,2,7,6] => [7,3,5,4,6,1,2] => ? = 1
[[[.,.],[[.,.],[.,.]]],[.,.]]
=> [1,3,5,4,2,7,6] => [1,4,5,3,2,7,6] => [7,4,3,5,6,1,2] => ? = 0
[[[.,.],[[.,[.,.]],.]],[.,.]]
=> [1,4,3,5,2,7,6] => [1,3,5,4,2,7,6] => [7,5,3,4,6,1,2] => ? = 0
[[[.,.],[[[.,.],.],.]],[.,.]]
=> [1,3,4,5,2,7,6] => [1,5,4,3,2,7,6] => [7,3,4,5,6,1,2] => ? = 0
[[[[.,.],.],[.,[.,.]]],[.,.]]
=> [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 0
[[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 0
[[[[.,.],[.,[.,.]]],.],[.,.]]
=> [1,4,3,2,5,7,6] => [1,3,4,2,5,7,6] => [7,5,4,6,3,1,2] => ? = 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: St000123
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●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
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [1,7,6,2,3,5,4] => [1,6,7,3,4,2,5] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [1,7,6,2,3,4,5] => [1,6,7,3,4,5,2] => ? = 3
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,3,7,6,5,4,2] => [1,7,2,6,5,4,3] => [1,5,2,6,7,3,4] => ? = 0
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => [1,5,2,6,7,4,3] => ? = 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => [1,7,2,6,3,5,4] => [1,6,2,7,4,3,5] => ? = 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => [1,6,2,7,3,5,4] => ? = 1
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,2,7,4,5,3] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => [1,3,6,2,7,4,5] => ? = 0
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => [1,7,3,2,6,4,5] => [1,3,6,2,7,5,4] => ? = 1
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [1,7,2,3,6,4,5] => [1,6,2,3,7,5,4] => ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => [1,7,4,2,3,6,5] => [1,4,6,3,2,7,5] => ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => [1,7,2,4,3,6,5] => [1,4,2,6,3,7,5] => ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => [1,3,6,2,4,7,5] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [1,7,5,2,3,4,6] => [1,5,7,3,4,2,6] => ? = 2
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [1,7,2,5,3,4,6] => [1,5,2,7,4,3,6] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => [1,3,5,2,7,4,6] => ? = 0
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,3,4,6,5,7,2] => [1,7,2,3,5,4,6] => [1,5,2,3,7,4,6] => ? = 0
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [1,7,4,2,3,5,6] => [1,4,7,3,2,5,6] => ? = 1
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,3,5,4,6,7,2] => [1,7,2,4,3,5,6] => [1,4,2,7,3,5,6] => ? = 0
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,5,6,7,3,4] => ? = 0
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [1,2,6,7,5,4,3] => [1,2,7,6,5,3,4] => [1,2,5,6,7,4,3] => ? = 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,2,5,7,6,4,3] => [1,2,7,6,3,5,4] => [1,2,6,7,4,3,5] => ? = 1
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,2,6,5,7,4,3] => [1,2,7,6,4,3,5] => [1,2,6,7,3,5,4] => ? = 1
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,2,5,6,7,4,3] => [1,2,7,6,3,4,5] => [1,2,6,7,4,5,3] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,2,6,3,7,4,5] => ? = 0
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => [1,2,6,3,7,5,4] => ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,2,5,4,7,6,3] => [1,2,7,4,3,6,5] => [1,2,4,6,3,7,5] => ? = 0
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [1,2,6,5,4,7,3] => [1,2,7,5,4,3,6] => [1,2,5,7,3,4,6] => ? = 0
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [1,2,5,6,4,7,3] => [1,2,7,5,3,4,6] => [1,2,5,7,4,3,6] => ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,2,5,3,7,4,6] => ? = 0
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,2,5,4,6,7,3] => [1,2,7,4,3,5,6] => [1,2,4,7,3,5,6] => ? = 0
[[[.,.],.],[[[[.,.],.],.],.]]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 0
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => [1,3,2,6,7,4,5] => ? = 0
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> [1,3,2,6,7,5,4] => [1,3,2,7,6,4,5] => [1,3,2,6,7,5,4] => ? = 1
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,3,2,5,7,6,4] => [1,3,2,7,4,6,5] => [1,3,2,6,4,7,5] => ? = 0
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [1,3,2,6,5,7,4] => [1,3,2,7,5,4,6] => [1,3,2,5,7,4,6] => ? = 0
[[[.,.],[.,.]],[[[.,.],.],.]]
=> [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => ? = 0
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,6,7,4,5] => ? = 0
[[[[.,.],.],.],[.,[[.,.],.]]]
=> [1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => ? = 1
[[[[.,.],.],.],[[.,[.,.]],.]]
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ? = 0
[[[[.,.],.],.],[[[.,.],.],.]]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 0
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => [1,3,4,2,6,7,5] => ? = 0
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,4,3,2,6,7,5] => [1,4,3,2,7,5,6] => [1,3,4,2,7,5,6] => ? = 0
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> [1,3,4,2,7,6,5] => [1,4,2,3,7,6,5] => [1,4,2,3,6,7,5] => ? = 0
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [1,3,4,2,6,7,5] => [1,4,2,3,7,5,6] => [1,4,2,3,7,5,6] => ? = 0
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 0
[[[[.,.],.],[.,.]],[[.,.],.]]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 0
[[[[.,.],[.,.]],.],[.,[.,.]]]
=> [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => ? = 0
[[[[.,.],[.,.]],.],[[.,.],.]]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => ? = 0
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 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: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●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
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [1,7,6,2,3,5,4] => [1,6,7,3,4,2,5] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [1,7,6,2,3,4,5] => [1,6,7,3,4,5,2] => ? = 3
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,3,7,6,5,4,2] => [1,7,2,6,5,4,3] => [1,5,2,6,7,3,4] => ? = 0
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => [1,5,2,6,7,4,3] => ? = 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => [1,7,2,6,3,5,4] => [1,6,2,7,4,3,5] => ? = 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => [1,6,2,7,3,5,4] => ? = 1
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,2,7,4,5,3] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => [1,3,6,2,7,4,5] => ? = 0
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => [1,7,3,2,6,4,5] => [1,3,6,2,7,5,4] => ? = 1
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [1,7,2,3,6,4,5] => [1,6,2,3,7,5,4] => ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => [1,7,4,2,3,6,5] => [1,4,6,3,2,7,5] => ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => [1,7,2,4,3,6,5] => [1,4,2,6,3,7,5] => ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => [1,3,6,2,4,7,5] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [1,7,5,2,3,4,6] => [1,5,7,3,4,2,6] => ? = 2
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [1,7,2,5,3,4,6] => [1,5,2,7,4,3,6] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => [1,3,5,2,7,4,6] => ? = 0
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,3,4,6,5,7,2] => [1,7,2,3,5,4,6] => [1,5,2,3,7,4,6] => ? = 0
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [1,7,4,2,3,5,6] => [1,4,7,3,2,5,6] => ? = 1
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,3,5,4,6,7,2] => [1,7,2,4,3,5,6] => [1,4,2,7,3,5,6] => ? = 0
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,5,6,7,3,4] => ? = 0
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [1,2,6,7,5,4,3] => [1,2,7,6,5,3,4] => [1,2,5,6,7,4,3] => ? = 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,2,5,7,6,4,3] => [1,2,7,6,3,5,4] => [1,2,6,7,4,3,5] => ? = 1
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,2,6,5,7,4,3] => [1,2,7,6,4,3,5] => [1,2,6,7,3,5,4] => ? = 1
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,2,5,6,7,4,3] => [1,2,7,6,3,4,5] => [1,2,6,7,4,5,3] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,2,6,3,7,4,5] => ? = 0
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => [1,2,6,3,7,5,4] => ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,2,5,4,7,6,3] => [1,2,7,4,3,6,5] => [1,2,4,6,3,7,5] => ? = 0
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [1,2,6,5,4,7,3] => [1,2,7,5,4,3,6] => [1,2,5,7,3,4,6] => ? = 0
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [1,2,5,6,4,7,3] => [1,2,7,5,3,4,6] => [1,2,5,7,4,3,6] => ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,2,5,3,7,4,6] => ? = 0
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,2,5,4,6,7,3] => [1,2,7,4,3,5,6] => [1,2,4,7,3,5,6] => ? = 0
[[[.,.],.],[[[[.,.],.],.],.]]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 0
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => [1,3,2,6,7,4,5] => ? = 0
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> [1,3,2,6,7,5,4] => [1,3,2,7,6,4,5] => [1,3,2,6,7,5,4] => ? = 1
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,3,2,5,7,6,4] => [1,3,2,7,4,6,5] => [1,3,2,6,4,7,5] => ? = 0
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [1,3,2,6,5,7,4] => [1,3,2,7,5,4,6] => [1,3,2,5,7,4,6] => ? = 0
[[[.,.],[.,.]],[[[.,.],.],.]]
=> [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => ? = 0
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,6,7,4,5] => ? = 0
[[[[.,.],.],.],[.,[[.,.],.]]]
=> [1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => ? = 1
[[[[.,.],.],.],[[.,[.,.]],.]]
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ? = 0
[[[[.,.],.],.],[[[.,.],.],.]]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 0
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => [1,3,4,2,6,7,5] => ? = 0
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,4,3,2,6,7,5] => [1,4,3,2,7,5,6] => [1,3,4,2,7,5,6] => ? = 0
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> [1,3,4,2,7,6,5] => [1,4,2,3,7,6,5] => [1,4,2,3,6,7,5] => ? = 0
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [1,3,4,2,6,7,5] => [1,4,2,3,7,5,6] => [1,4,2,3,7,5,6] => ? = 0
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 0
[[[[.,.],.],[.,.]],[[.,.],.]]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 0
[[[[.,.],[.,.]],.],[.,[.,.]]]
=> [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => ? = 0
[[[[.,.],[.,.]],.],[[.,.],.]]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => ? = 0
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 0
Description
The number of nestings in the permutation.
Matching statistic: St000039
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●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
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,4,5,7,6,3,2] => [1,7,6,2,3,5,4] => ? = 2
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => [1,7,6,2,3,4,5] => ? = 3
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,3,7,6,5,4,2] => [1,7,2,6,5,4,3] => ? = 0
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => ? = 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,3,5,7,6,4,2] => [1,7,2,6,3,5,4] => ? = 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => ? = 1
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => ? = 2
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => ? = 0
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => [1,7,3,2,6,4,5] => ? = 1
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,3,4,7,6,5,2] => [1,7,2,3,6,5,4] => ? = 0
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,3,4,6,7,5,2] => [1,7,2,3,6,4,5] => ? = 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => [1,7,4,2,3,6,5] => ? = 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => [1,7,2,4,3,6,5] => ? = 0
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => ? = 0
[[.,.],[[[[.,.],.],.],[.,.]]]
=> [1,3,4,5,7,6,2] => [1,7,2,3,4,6,5] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [1,4,5,6,3,7,2] => [1,7,5,2,3,4,6] => ? = 2
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [1,3,6,5,4,7,2] => [1,7,2,5,4,3,6] => ? = 0
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,3,5,6,4,7,2] => [1,7,2,5,3,4,6] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 0
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,3,4,6,5,7,2] => [1,7,2,3,5,4,6] => ? = 0
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [1,4,5,3,6,7,2] => [1,7,4,2,3,5,6] => ? = 1
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,3,5,4,6,7,2] => [1,7,2,4,3,5,6] => ? = 0
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 0
[[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 0
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 0
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [1,2,6,7,5,4,3] => [1,2,7,6,5,3,4] => ? = 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [1,2,5,7,6,4,3] => [1,2,7,6,3,5,4] => ? = 1
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,2,6,5,7,4,3] => [1,2,7,6,4,3,5] => ? = 1
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [1,2,5,6,7,4,3] => [1,2,7,6,3,4,5] => ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => ? = 0
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,2,5,4,7,6,3] => [1,2,7,4,3,6,5] => ? = 0
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 0
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [1,2,6,5,4,7,3] => [1,2,7,5,4,3,6] => ? = 0
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [1,2,5,6,4,7,3] => [1,2,7,5,3,4,6] => ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 0
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,2,5,4,6,7,3] => [1,2,7,4,3,5,6] => ? = 0
[[[.,.],.],[[[[.,.],.],.],.]]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => ? = 0
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> [1,3,2,6,7,5,4] => [1,3,2,7,6,4,5] => ? = 1
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,3,2,5,7,6,4] => [1,3,2,7,4,6,5] => ? = 0
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [1,3,2,6,5,7,4] => [1,3,2,7,5,4,6] => ? = 0
[[[.,.],[.,.]],[[[.,.],.],.]]
=> [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => ? = 0
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 0
[[[[.,.],.],.],[.,[[.,.],.]]]
=> [1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => ? = 1
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 0
[[[[.,.],.],.],[[.,[.,.]],.]]
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 0
[[[[.,.],.],.],[[[.,.],.],.]]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,4,3,2,7,6,5] => [1,4,3,2,7,6,5] => ? = 0
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,4,3,2,6,7,5] => [1,4,3,2,7,5,6] => ? = 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: St001330
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 30%●distinct values known / distinct values provided: 25%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 30%●distinct values known / distinct values provided: 25%
Values
[.,.]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[.,.],.]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[.,[.,[[[[.,.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[[[.,.],[.,.]],[.,[.,.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,.],[.,[.,.]]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],.],[.,.]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[.,.],[.,.]],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[[.,[.,.]],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],[.,[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001906
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 75%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 75%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ? = 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ? = 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 3
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? = 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? = 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 2
[.,[.,[[[[.,.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 3
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? = 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? = 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ? = 0
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ? = 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? = 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ? = 1
[.,[[.,[[[.,.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? = 0
[.,[[[.,.],[[.,.],.]],.]]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ? = 1
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 0
[.,[[[[.,.],.],[.,.]],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ? = 0
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 0
[.,[[[.,[[.,.],.]],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 1
[.,[[[[.,.],[.,.]],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 0
[.,[[[[.,[.,.]],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 1
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 1
[[.,.],[.,[[[.,.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ? = 2
[[.,.],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 0
[[.,.],[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? = 0
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001811The Castelnuovo-Mumford regularity of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001862The number of crossings of a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St001868The number of alignments of type NE of a signed permutation. St000629The defect of a binary word. St000805The number of peaks of the associated bargraph. 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 $K[x]/(x^n)$. St000445The number of rises of length 1 of a Dyck 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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