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Matching statistic: St001465
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St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000237
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 86%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 86%
Values
[1] => {{1}}
=> [1] => [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
[3,1,2] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 0
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 0
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 0
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => 0
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 0
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => 0
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => 0
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => 0
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => 0
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => 0
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 0
[1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 0
[1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 0
[1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => {{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 0
[1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,5,7,4,6] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,5,7,6,4] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 0
[1,2,3,6,4,5,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 0
[1,2,3,6,4,7,5] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 0
[1,2,3,6,5,7,4] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 0
[1,2,3,6,7,4,5] => {{1},{2},{3},{4,6},{5,7}}
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ? = 0
[1,2,3,6,7,5,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,4,5,6] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,4,6,5] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 0
[1,2,3,7,5,4,6] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 0
[1,2,3,7,5,6,4] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 0
[1,2,3,7,6,4,5] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,6,5,4] => {{1},{2},{3},{4,7},{5,6}}
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 1
[1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => {{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[1,2,4,3,6,5,7] => {{1},{2},{3,4},{5,6},{7}}
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2
[1,2,4,3,6,7,5] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,5,6] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,6,5] => {{1},{2},{3,4},{5,7},{6}}
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 1
[1,2,4,5,3,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 0
[1,2,4,5,3,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,2,4,5,6,3,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 0
[1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,5,7,3,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,5,7,6,3] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 0
[1,2,4,6,3,5,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 0
[1,2,4,6,3,7,5] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,6,5,3,7] => {{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => ? = 0
[1,2,4,6,5,7,3] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 0
[1,2,4,6,7,3,5] => {{1},{2},{3,4,6},{5,7}}
=> [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => ? = 0
[1,2,4,6,7,5,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,7,3,5,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,7,3,6,5] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 0
[1,2,4,7,5,3,6] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 0
[1,2,4,7,5,6,3] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => ? = 0
[1,2,4,7,6,3,5] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,7,6,5,3] => {{1},{2},{3,4,7},{5,6}}
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => ? = 1
[1,2,5,3,4,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 0
[1,2,5,3,4,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,2,5,3,6,4,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 0
[1,2,5,3,6,7,4] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,5,3,7,4,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000214
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 86%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 86%
Values
[1] => {{1}}
=> [1] => [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
[3,1,2] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 0
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 0
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 0
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 0
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 0
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 0
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 0
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 0
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 0
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => 0
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => 0
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => 0
[1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 0
[1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 0
[1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 0
[1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => {{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 0
[1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,5,7,4,6] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,5,7,6,4] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => ? = 0
[1,2,3,6,4,5,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 0
[1,2,3,6,4,7,5] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 0
[1,2,3,6,5,7,4] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => ? = 0
[1,2,3,6,7,5,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,4,5,6] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,4,6,5] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => ? = 0
[1,2,3,7,5,4,6] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => ? = 0
[1,2,3,7,5,6,4] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => ? = 0
[1,2,3,7,6,4,5] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,6,5,4] => {{1},{2},{3},{4,7},{5,6}}
=> [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => ? = 1
[1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => {{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[1,2,4,3,6,5,7] => {{1},{2},{3,4},{5,6},{7}}
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2
[1,2,4,3,6,7,5] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,5,6] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,6,5] => {{1},{2},{3,4},{5,7},{6}}
=> [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 1
[1,2,4,5,3,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 0
[1,2,4,5,3,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,2,4,5,6,3,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 0
[1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,5,7,3,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,5,7,6,3] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => ? = 0
[1,2,4,6,3,5,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 0
[1,2,4,6,3,7,5] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,6,5,3,7] => {{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ? = 0
[1,2,4,6,5,7,3] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => ? = 0
[1,2,4,6,7,5,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,7,3,5,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,7,3,6,5] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => ? = 0
[1,2,4,7,5,3,6] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => ? = 0
[1,2,4,7,5,6,3] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => ? = 0
[1,2,4,7,6,3,5] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,4,7,6,5,3] => {{1},{2},{3,4,7},{5,6}}
=> [1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => ? = 1
[1,2,5,3,4,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 0
[1,2,5,3,4,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,2,5,3,6,4,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 0
[1,2,5,3,6,7,4] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,5,3,7,4,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 0
[1,2,5,3,7,6,4] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => ? = 0
[1,2,5,4,3,6,7] => {{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ? = 0
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
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