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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000237
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(load all 5 compositions to match this statistic)
St000237: 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] => 2
[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] => 2
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[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] => 1
[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] => 2
[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] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000382
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 91%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 91%
Values
[1] => {{1}}
=> {{1}}
=> [1] => 1 = 0 + 1
[1,2] => {{1},{2}}
=> {{1},{2}}
=> [1,1] => 1 = 0 + 1
[2,1] => {{1,2}}
=> {{1,2}}
=> [2] => 2 = 1 + 1
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => 1 = 0 + 1
[1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => 2 = 1 + 1
[2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => 2 = 1 + 1
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> [3] => 3 = 2 + 1
[3,1,2] => {{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => 1 = 0 + 1
[3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => 1 = 0 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => 2 = 1 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => 2 = 1 + 1
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => 3 = 2 + 1
[1,4,2,3] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => 1 = 0 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => 1 = 0 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => 2 = 1 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => 3 = 2 + 1
[2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => 3 = 2 + 1
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => 4 = 3 + 1
[2,4,1,3] => {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => 2 = 1 + 1
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => 2 = 1 + 1
[3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => 1 = 0 + 1
[3,1,4,2] => {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => 2 = 1 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => 1 = 0 + 1
[3,2,4,1] => {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => 2 = 1 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => 1 = 0 + 1
[3,4,2,1] => {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => 1 = 0 + 1
[4,1,2,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1 = 0 + 1
[4,1,3,2] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1 = 0 + 1
[4,2,1,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1 = 0 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1 = 0 + 1
[4,3,1,2] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => 2 = 1 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => 2 = 1 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => 2 = 1 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => 2 = 1 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> [3,1,1] => 3 = 2 + 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 1 = 0 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 1 = 0 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => 2 = 1 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> [3,1,1] => 3 = 2 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1] => 3 = 2 + 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> [4,1] => 4 = 3 + 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> [2,2,1] => 2 = 1 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> [2,2,1] => 2 = 1 + 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 1 = 0 + 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> [2,2,1] => 2 = 1 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 1 = 0 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> [2,2,1] => 2 = 1 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1},{2,4,5},{3}}
=> [1,3,1] => 1 = 0 + 1
[7,8,6,4,5,3,2,1] => {{1,7},{2,8},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,6,7,4,5,3,2,1] => {{1,8},{2,6},{3,7},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[6,7,8,3,4,5,2,1] => {{1,6},{2,7},{3,8},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,7,6,4,5,2,3,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[5,4,6,3,7,2,8,1] => {{1,5,7,8},{2,4},{3,6}}
=> {{1,8},{2,4,7},{3,5,6}}
=> ? => ? = 1 + 1
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? => ? = 4 + 1
[8,7,6,4,5,3,1,2] => {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,7,6,4,5,1,2,3] => {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,6,7,4,5,1,2,3] => {{1,8},{2,6},{3,7},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,7,4,5,6,1,2,3] => {{1,8},{2,7},{3,4,5,6}}
=> ?
=> ? => ? = 3 + 1
[7,8,4,5,6,1,2,3] => {{1,7},{2,8},{3,4,5,6}}
=> ?
=> ? => ? = 3 + 1
[6,7,4,5,8,1,2,3] => {{1,6},{2,7},{3,4,5,8}}
=> ?
=> ? => ? = 2 + 1
[8,7,6,4,3,2,1,5] => {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,7,6,4,2,1,3,5] => {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,7,6,1,2,3,4,5] => {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[8,6,7,1,2,3,4,5] => {{1,8},{2,6},{3,7},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[6,7,8,1,2,3,4,5] => {{1,6},{2,7},{3,8},{4},{5}}
=> ?
=> ? => ? = 0 + 1
[7,8,5,4,2,1,3,6] => {{1,7},{2,8},{3,5},{4},{6}}
=> ?
=> ? => ? = 0 + 1
[7,8,5,4,1,2,3,6] => {{1,7},{2,8},{3,5},{4},{6}}
=> ?
=> ? => ? = 0 + 1
[8,6,5,4,3,2,1,7] => {{1,8},{2,6},{3,5},{4},{7}}
=> ?
=> ? => ? = 0 + 1
[8,6,5,4,3,1,2,7] => {{1,8},{2,6},{3,5},{4},{7}}
=> ?
=> ? => ? = 0 + 1
[8,5,6,3,4,1,2,7] => {{1,8},{2,5},{3,6},{4},{7}}
=> ?
=> ? => ? = 0 + 1
[8,6,5,4,2,1,3,7] => {{1,8},{2,6},{3,5},{4},{7}}
=> ?
=> ? => ? = 0 + 1
[8,6,4,3,2,1,5,7] => {{1,8},{2,6},{3,4},{5},{7}}
=> {{1,4},{2},{3,6},{5},{7,8}}
=> ? => ? = 1 + 1
[8,6,4,2,1,3,5,7] => {{1,8},{2,6},{3,4},{5},{7}}
=> {{1,4},{2},{3,6},{5},{7,8}}
=> ? => ? = 1 + 1
[6,7,5,4,3,2,1,8] => {{1,6},{2,7},{3,5},{4},{8}}
=> ?
=> ? => ? = 0 + 1
[6,5,7,4,3,2,1,8] => {{1,6},{2,5},{3,7},{4},{8}}
=> ?
=> ? => ? = 0 + 1
[5,6,7,4,3,2,1,8] => {{1,5},{2,6},{3,7},{4},{8}}
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? => ? = 0 + 1
[7,6,4,5,3,2,1,8] => ?
=> ?
=> ? => ? = 2 + 1
[6,5,4,7,3,2,1,8] => {{1,6},{2,5},{3,4,7},{8}}
=> {{1,4},{2,5},{3,6,7},{8}}
=> ? => ? = 1 + 1
[7,5,6,3,4,2,1,8] => {{1,7},{2,5},{3,6},{4},{8}}
=> {{1},{2},{3,5,6},{4,7},{8}}
=> ? => ? = 0 + 1
[7,6,4,3,5,2,1,8] => ?
=> ?
=> ? => ? = 1 + 1
[7,6,3,4,5,2,1,8] => ?
=> ?
=> ? => ? = 0 + 1
[7,5,4,3,6,2,1,8] => ?
=> ?
=> ? => ? = 2 + 1
[6,7,5,4,2,3,1,8] => {{1,6},{2,7},{3,5},{4},{8}}
=> ?
=> ? => ? = 0 + 1
[6,5,7,3,2,4,1,8] => {{1,6},{2,5},{3,7},{4},{8}}
=> ?
=> ? => ? = 0 + 1
[5,6,7,2,3,4,1,8] => {{1,5},{2,6},{3,7},{4},{8}}
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? => ? = 0 + 1
[5,6,7,1,2,3,4,8] => {{1,5},{2,6},{3,7},{4},{8}}
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? => ? = 0 + 1
[6,7,1,2,3,4,5,8] => {{1,6},{2,7},{3},{4},{5},{8}}
=> ?
=> ? => ? = 0 + 1
[7,1,2,3,4,5,6,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? => ? = 0 + 1
[6,5,4,3,2,1,7,8] => {{1,6},{2,5},{3,4},{7},{8}}
=> ?
=> ? => ? = 1 + 1
[5,6,3,4,1,2,7,8] => {{1,5},{2,6},{3},{4},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? => ? = 0 + 1
[4,5,6,1,2,3,7,8] => {{1,4},{2,5},{3,6},{7},{8}}
=> {{1},{2},{3,4,5,6},{7},{8}}
=> ? => ? = 0 + 1
[5,4,3,2,1,6,7,8] => {{1,5},{2,4},{3},{6},{7},{8}}
=> {{1},{2,4},{3,5},{6},{7},{8}}
=> ? => ? = 0 + 1
[5,1,2,3,4,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? => ? = 0 + 1
[3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? => ? = 0 + 1
[3,5,8,7,6,4,2,1] => {{1,3,8},{2,5,6},{4,7}}
=> ?
=> ? => ? = 1 + 1
[3,2,6,5,7,4,8,1] => {{1,3,6},{2},{4,5,7,8}}
=> {{1,5,8},{2,3,6,7},{4}}
=> ? => ? = 2 + 1
[4,3,5,2,7,6,8,1] => {{1,4},{2,3,5,7,8},{6}}
=> {{1,3,8},{2,4,5,7},{6}}
=> ? => ? = 2 + 1
[1,7,8,6,5,4,3,2] => {{1},{2,7},{3,8},{4,6},{5}}
=> ?
=> ? => ? = 0 + 1
Description
The first part of an integer composition.
Matching statistic: St000932
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 73%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[3,1,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,4,2,3] => [3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,4,3,2] => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[2,4,1,3] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[4,1,3,2] => [4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,1,3] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,4,3] => [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,3,2,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [5,2,4,3,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [3,4,2,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [4,5,2,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,4,5,3,2] => [5,4,2,3,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,2,1,3,8] => [4,7,3,5,2,6,1,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,5,6,4,3,2,1] => [7,6,5,3,4,1,2,8] => [5,4,3,7,2,6,1,8] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 0
[7,8,6,4,5,3,2,1] => [7,6,4,5,3,1,2,8] => [5,3,4,7,2,6,1,8] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 0
[8,6,7,4,5,3,2,1] => [7,6,4,5,2,3,1,8] => [6,3,4,5,2,7,1,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [7,6,4,3,2,1,5,8] => [4,3,7,5,2,6,1,8] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[6,5,7,4,8,3,2,1] => [7,6,4,2,1,3,5,8] => [6,3,4,2,7,5,1,8] => ?
=> ? = 0
[7,8,6,5,3,4,2,1] => [7,5,6,4,3,1,2,8] => [4,7,2,5,3,6,1,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,3,4,2,1] => [7,5,6,4,2,3,1,8] => [4,5,2,6,3,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1,8] => [6,2,5,4,3,7,1,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[6,7,8,3,4,5,2,1] => [7,4,5,6,1,2,3,8] => [6,2,4,7,3,5,1,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[7,8,6,5,4,2,3,1] => [6,7,5,4,3,1,2,8] => [4,5,3,6,1,7,2,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1,8] => [4,7,1,6,3,5,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,7,5,6,4,2,3,1] => [6,7,5,3,4,2,1,8] => [5,4,3,7,1,6,2,8] => ?
=> ? = 0
[8,7,6,4,5,2,3,1] => [6,7,4,5,3,2,1,8] => [5,3,4,7,1,6,2,8] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 0
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1,8] => [5,2,7,1,6,4,3,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0
[6,7,8,5,2,3,4,1] => [5,6,7,4,1,2,3,8] => [4,5,1,6,2,7,3,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1,8] => [6,4,2,7,1,5,3,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[7,8,3,2,4,5,6,1] => [4,3,5,6,7,1,2,8] => [6,1,4,7,2,3,5,8] => ?
=> ? = 0
[7,6,5,4,3,2,8,1] => [6,5,4,3,2,1,7,8] => [4,3,5,2,6,1,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1
[6,5,7,4,3,2,8,1] => [6,5,4,2,1,3,7,8] => [6,3,4,2,5,1,7,8] => ?
=> ? = 1
[6,5,4,3,7,2,8,1] => [6,4,3,2,1,5,7,8] => [3,4,2,6,5,1,7,8] => ?
=> ? = 2
[5,4,6,3,7,2,8,1] => [6,4,2,1,3,5,7,8] => [6,5,3,2,4,1,7,8] => ?
=> ? = 1
[3,4,5,2,6,7,8,1] => [4,1,2,3,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[5,2,3,4,6,7,8,1] => [2,3,4,1,5,6,7,8] => [4,1,2,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,2,5,6,7,8,1] => [3,1,2,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[4,2,3,5,6,7,8,1] => [2,3,1,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[8,7,6,5,4,3,1,2] => [8,6,5,4,3,2,1,7] => [4,5,3,6,2,8,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[7,8,6,5,4,3,1,2] => [8,6,5,4,3,1,2,7] => [4,5,3,8,7,2,6,1] => [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 1
[8,6,7,5,4,3,1,2] => [8,6,5,4,2,3,1,7] => [4,6,3,5,2,8,7,1] => [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[8,7,5,6,4,3,1,2] => [8,6,5,3,4,2,1,7] => [5,4,3,6,2,8,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 0
[8,7,6,4,5,3,1,2] => [8,6,4,5,3,2,1,7] => [5,3,4,6,2,8,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 0
[8,7,6,5,3,4,1,2] => [8,5,6,4,3,2,1,7] => [4,6,2,5,3,8,7,1] => [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[8,7,5,6,3,4,1,2] => [8,5,6,3,4,2,1,7] => [6,2,5,4,3,8,7,1] => ?
=> ? = 0
[7,8,5,6,3,4,1,2] => [8,5,6,3,4,1,2,7] => [8,7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [8,5,6,1,2,3,4,7] => [5,2,6,3,8,7,4,1] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [8,3,4,5,6,1,2,7] => [8,7,2,3,4,5,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,4,5,6,7,8,1,2] => [8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [6,8,5,4,3,2,1,7] => [4,5,3,8,7,1,6,2] => ?
=> ? = 1
[7,6,8,5,4,2,1,3] => [6,8,5,4,2,1,3,7] => [4,6,1,8,7,3,5,2] => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 1
[8,7,6,5,4,1,2,3] => [7,8,5,4,3,2,1,6] => [4,5,3,7,1,8,6,2] => ?
=> ? = 1
[6,7,8,5,4,1,2,3] => [7,8,5,4,1,2,3,6] => [4,7,3,5,1,8,6,2] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1
[8,5,6,7,4,1,2,3] => [7,8,5,2,3,4,1,6] => [5,3,7,1,8,6,4,2] => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 0
[8,7,6,4,5,1,2,3] => [7,8,4,5,3,2,1,6] => [5,3,4,7,1,8,6,2] => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> ? = 0
[8,6,7,4,5,1,2,3] => [7,8,4,5,2,3,1,6] => [7,1,8,6,3,4,5,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[8,7,4,5,6,1,2,3] => [7,8,3,4,5,2,1,6] => [3,4,5,7,1,8,6,2] => ?
=> ? = 3
[7,8,4,5,6,1,2,3] => [7,8,3,4,5,1,2,6] => [3,4,5,8,6,1,7,2] => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[6,7,4,5,8,1,2,3] => [7,8,3,4,1,2,5,6] => [3,4,7,5,1,8,6,2] => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[7,6,5,8,3,2,1,4] => [6,5,8,3,2,1,4,7] => [5,2,6,1,8,7,4,3] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000214
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 91%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,2,1] => 2
[3,1,2] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 3
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 0
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 0
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => [1,6,3,7,4,2,5] => ? = 0
[1,4,6,7,5,3,2] => [1,7,6,2,5,3,4] => [1,5,6,3,7,4,2] => ? = 0
[1,4,7,2,3,5,6] => [1,4,5,2,6,7,3] => [1,4,2,7,3,5,6] => ? = 0
[1,4,7,6,5,3,2] => [1,7,6,2,5,4,3] => [1,5,7,3,6,4,2] => ? = 0
[1,5,2,7,3,4,6] => [1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => ? = 0
[1,5,6,2,7,3,4] => [1,4,6,7,2,3,5] => [1,6,3,7,5,2,4] => ? = 0
[1,5,6,4,7,3,2] => [1,7,6,4,2,3,5] => [1,4,6,3,7,5,2] => ? = 0
[1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => [1,5,2,6,3,7,4] => ? = 0
[1,5,6,7,4,3,2] => [1,7,6,5,2,3,4] => [1,6,3,7,4,5,2] => ? = 0
[1,5,7,2,3,4,6] => [1,4,5,6,2,7,3] => [1,7,3,5,2,4,6] => ? = 0
[1,5,7,6,4,3,2] => [1,7,6,5,2,4,3] => [1,7,3,6,4,5,2] => ? = 0
[1,6,2,3,4,7,5] => [1,3,4,5,7,2,6] => [1,7,6,2,3,4,5] => ? = 1
[1,6,2,7,3,4,5] => [1,3,5,6,7,2,4] => [1,7,4,6,2,3,5] => ? = 0
[1,6,5,4,3,7,2] => [1,7,5,4,3,2,6] => [1,4,5,3,7,6,2] => ? = 1
[1,6,5,4,7,3,2] => [1,7,6,4,3,2,5] => [1,4,7,5,3,6,2] => ? = 0
[1,6,7,2,3,4,5] => [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => ? = 0
[1,6,7,5,4,3,2] => [1,7,6,5,4,2,3] => [1,5,4,7,3,6,2] => ? = 1
[1,7,3,5,6,4,2] => [1,7,3,6,4,5,2] => [1,3,6,5,4,7,2] => ? = 2
[1,7,3,6,5,4,2] => [1,7,3,6,5,4,2] => [1,3,5,6,4,7,2] => ? = 0
[1,7,4,3,6,5,2] => [1,7,4,3,6,5,2] => [1,4,3,6,5,7,2] => ? = 2
[1,7,4,5,3,6,2] => [1,7,5,3,4,6,2] => [1,5,4,3,6,7,2] => ? = 2
[1,7,4,6,5,3,2] => [1,7,6,3,5,4,2] => [1,5,6,4,3,7,2] => ? = 1
[1,7,5,4,3,6,2] => [1,7,5,4,3,6,2] => [1,4,5,3,6,7,2] => ? = 0
[1,7,5,4,6,3,2] => [1,7,6,4,3,5,2] => [1,4,6,5,3,7,2] => ? = 1
[1,7,6,4,5,3,2] => [1,7,6,4,5,3,2] => [1,4,5,6,3,7,2] => ? = 0
[1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => ? = 1
[2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => ? = 1
[2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 4
[2,3,4,5,7,1,6] => [6,1,2,3,4,7,5] => [7,5,4,3,2,1,6] => ? = 4
[2,3,4,6,1,7,5] => [5,1,2,3,7,4,6] => [7,6,4,3,2,1,5] => ? = 4
[2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [6,4,3,2,1,7,5] => ? = 3
[2,3,4,7,1,5,6] => [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => ? = 3
[2,3,5,6,7,1,4] => [6,1,2,7,3,4,5] => [7,5,3,2,1,6,4] => ? = 2
[2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => [4,2,1,3,5,6,7] => ? = 1
[2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => [5,2,1,3,4,6,7] => ? = 1
[2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => [7,3,5,2,1,4,6] => ? = 1
[2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => [6,2,1,3,4,5,7] => ? = 1
[2,6,1,7,3,4,5] => [3,1,5,6,7,2,4] => [7,4,6,2,1,3,5] => ? = 1
[2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => [6,2,1,4,7,3,5] => ? = 1
[3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => [2,7,6,5,4,3,1] => ? = 4
[3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 0
[3,4,2,5,6,7,1] => [7,3,1,2,4,5,6] => [7,6,5,4,2,3,1] => ? = 3
[3,4,5,2,6,7,1] => [7,4,1,2,3,5,6] => [4,2,7,6,5,3,1] => ? = 2
[3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => [7,6,4,2,5,3,1] => ? = 1
[3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => [7,5,3,1,6,4,2] => ? = 0
[3,5,6,1,2,4,7] => [4,5,1,6,2,3,7] => [5,2,6,3,1,4,7] => ? = 0
[3,5,6,4,2,1,7] => [6,5,1,4,2,3,7] => [4,5,2,6,3,1,7] => ? = 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].
Matching statistic: St000502
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
St000502: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
Values
[1] => {{1}}
=> ? = 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => {{1,3},{2}}
=> 0
[3,2,1] => {{1,3},{2}}
=> 0
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> 0
[1,4,3,2] => {{1},{2,4},{3}}
=> 0
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 3
[2,4,1,3] => {{1,2,4},{3}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> 1
[3,1,2,4] => {{1,3},{2},{4}}
=> 0
[3,1,4,2] => {{1,3,4},{2}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 0
[3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> 0
[3,4,2,1] => {{1,3},{2,4}}
=> 0
[4,1,2,3] => {{1,4},{2},{3}}
=> 0
[4,1,3,2] => {{1,4},{2},{3}}
=> 0
[4,2,1,3] => {{1,4},{2},{3}}
=> 0
[4,2,3,1] => {{1,4},{2},{3}}
=> 0
[4,3,1,2] => {{1,4},{2,3}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 0
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 0
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1
[7,6,8,5,4,3,2,1] => {{1,7},{2,6},{3,8},{4,5}}
=> ? = 1
[7,8,5,6,4,3,2,1] => {{1,7},{2,8},{3,5},{4,6}}
=> ? = 0
[7,8,6,4,5,3,2,1] => {{1,7},{2,8},{3,6},{4},{5}}
=> ? = 0
[8,6,7,4,5,3,2,1] => {{1,8},{2,6},{3,7},{4},{5}}
=> ? = 0
[7,6,5,4,8,3,2,1] => {{1,7},{2,6},{3,5,8},{4}}
=> ? = 0
[6,5,7,4,8,3,2,1] => {{1,6},{2,5,8},{3,7},{4}}
=> ? = 0
[7,8,6,5,3,4,2,1] => {{1,7},{2,8},{3,6},{4,5}}
=> ? = 1
[8,6,7,5,3,4,2,1] => {{1,8},{2,6},{3,7},{4,5}}
=> ? = 1
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> ? = 0
[6,7,8,3,4,5,2,1] => {{1,6},{2,7},{3,8},{4},{5}}
=> ? = 0
[7,8,6,5,4,2,3,1] => {{1,7},{2,8},{3,6},{4,5}}
=> ? = 1
[8,6,7,5,4,2,3,1] => {{1,8},{2,6},{3,7},{4,5}}
=> ? = 1
[8,7,5,6,4,2,3,1] => {{1,8},{2,7},{3,5},{4,6}}
=> ? = 0
[8,7,6,4,5,2,3,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? = 0
[8,6,5,7,3,2,4,1] => {{1,8},{2,6},{3,5},{4,7}}
=> ? = 0
[6,7,8,5,2,3,4,1] => {{1,6},{2,7},{3,8},{4,5}}
=> ? = 1
[8,5,6,7,2,3,4,1] => {{1,8},{2,5},{3,6},{4,7}}
=> ? = 0
[7,8,3,2,4,5,6,1] => {{1,7},{2,8},{3},{4},{5},{6}}
=> ? = 0
[7,6,5,4,3,2,8,1] => {{1,7,8},{2,6},{3,5},{4}}
=> ? = 1
[6,5,7,4,3,2,8,1] => {{1,6},{2,5},{3,7,8},{4}}
=> ? = 1
[6,5,4,3,7,2,8,1] => {{1,6},{2,5,7,8},{3,4}}
=> ? = 2
[5,4,6,3,7,2,8,1] => {{1,5,7,8},{2,4},{3,6}}
=> ? = 1
[5,2,3,4,6,7,8,1] => {{1,5,6,7,8},{2},{3},{4}}
=> ? = 3
[8,7,6,5,4,3,1,2] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1
[7,8,6,5,4,3,1,2] => {{1,7},{2,8},{3,6},{4,5}}
=> ? = 1
[8,6,7,5,4,3,1,2] => {{1,8},{2,6},{3,7},{4,5}}
=> ? = 1
[8,7,5,6,4,3,1,2] => {{1,8},{2,7},{3,5},{4,6}}
=> ? = 0
[8,7,6,4,5,3,1,2] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? = 0
[8,7,6,5,3,4,1,2] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1
[8,7,5,6,3,4,1,2] => {{1,8},{2,7},{3,5},{4,6}}
=> ? = 0
[7,8,5,6,3,4,1,2] => {{1,7},{2,8},{3,5},{4,6}}
=> ? = 0
[5,6,7,8,3,4,1,2] => {{1,5},{2,6},{3,7},{4,8}}
=> ? = 0
[7,8,3,4,5,6,1,2] => {{1,7},{2,8},{3},{4},{5},{6}}
=> ? = 0
[3,4,5,6,7,8,1,2] => {{1,3,5,7},{2,4,6,8}}
=> ? = 0
[8,7,6,5,4,2,1,3] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1
[7,6,8,5,4,2,1,3] => {{1,7},{2,6},{3,8},{4,5}}
=> ? = 1
[8,7,6,5,4,1,2,3] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1
[6,7,8,5,4,1,2,3] => {{1,6},{2,7},{3,8},{4,5}}
=> ? = 1
[8,5,6,7,4,1,2,3] => {{1,8},{2,5},{3,6},{4,7}}
=> ? = 0
[8,7,6,4,5,1,2,3] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? = 0
[8,6,7,4,5,1,2,3] => {{1,8},{2,6},{3,7},{4},{5}}
=> ? = 0
[8,7,4,5,6,1,2,3] => {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[7,8,4,5,6,1,2,3] => {{1,7},{2,8},{3,4,5,6}}
=> ? = 3
[6,7,4,5,8,1,2,3] => {{1,6},{2,7},{3,4,5,8}}
=> ? = 2
[7,6,5,8,3,2,1,4] => {{1,7},{2,6},{3,5},{4,8}}
=> ? = 0
[7,5,6,8,2,3,1,4] => {{1,7},{2,5},{3,6},{4,8}}
=> ? = 0
[6,7,5,8,3,1,2,4] => {{1,6},{2,7},{3,5},{4,8}}
=> ? = 0
[6,5,7,8,2,1,3,4] => {{1,6},{2,5},{3,7},{4,8}}
=> ? = 0
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000504
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
Values
[1] => {{1}}
=> {{1}}
=> {{1}}
=> ? = 0 + 1
[1,2] => {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
[2,1] => {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 2 = 1 + 1
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[1,3,2] => {{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 2 = 1 + 1
[2,1,3] => {{1,2},{3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2 = 1 + 1
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 3 = 2 + 1
[3,1,2] => {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1 = 0 + 1
[3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1 = 0 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3 = 2 + 1
[1,4,2,3] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3 = 2 + 1
[2,3,1,4] => {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3 = 2 + 1
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4 = 3 + 1
[2,4,1,3] => {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[3,1,4,2] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[3,2,4,1] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[3,4,2,1] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[4,1,2,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[4,1,3,2] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[4,2,1,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[4,3,1,2] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2 = 1 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2 = 1 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 4 = 3 + 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 2 = 1 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 2 = 1 + 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1 = 0 + 1
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => {{1,7},{2,6},{3,8},{4,5}}
=> {{1,6},{2,8},{3,7},{4,5}}
=> {{1,5},{2},{3,7},{4,6,8}}
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => {{1,7},{2,8},{3,5},{4,6}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> ? = 0 + 1
[7,8,6,4,5,3,2,1] => {{1,7},{2,8},{3,6},{4},{5}}
=> {{1,7},{2,8},{3,6},{4},{5}}
=> ?
=> ? = 0 + 1
[8,6,7,4,5,3,2,1] => {{1,8},{2,6},{3,7},{4},{5}}
=> {{1,8},{2,6},{3,7},{4},{5}}
=> ?
=> ? = 0 + 1
[7,6,5,4,8,3,2,1] => {{1,7},{2,6},{3,5,8},{4}}
=> {{1,4,6},{2,8},{3,7},{5}}
=> {{1},{2,6},{3,4,7},{5,8}}
=> ? = 0 + 1
[6,5,7,4,8,3,2,1] => {{1,6},{2,5,8},{3,7},{4}}
=> {{1,4,7},{2,6},{3,8},{5}}
=> {{1},{2},{3,4,7},{5,6,8}}
=> ? = 0 + 1
[7,8,6,5,3,4,2,1] => {{1,7},{2,8},{3,6},{4,5}}
=> {{1,7},{2,8},{3,6},{4,5}}
=> {{1,5},{2,6},{3},{4,7,8}}
=> ? = 1 + 1
[8,6,7,5,3,4,2,1] => {{1,8},{2,6},{3,7},{4,5}}
=> {{1,8},{2,6},{3,7},{4,5}}
=> {{1,5},{2},{3,6,7},{4,8}}
=> ? = 1 + 1
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => {{1,6},{2,7},{3,8},{4},{5}}
=> {{1,6},{2,7},{3,8},{4},{5}}
=> ?
=> ? = 0 + 1
[7,8,6,5,4,2,3,1] => {{1,7},{2,8},{3,6},{4,5}}
=> {{1,7},{2,8},{3,6},{4,5}}
=> {{1,5},{2,6},{3},{4,7,8}}
=> ? = 1 + 1
[8,6,7,5,4,2,3,1] => {{1,8},{2,6},{3,7},{4,5}}
=> {{1,8},{2,6},{3,7},{4,5}}
=> {{1,5},{2},{3,6,7},{4,8}}
=> ? = 1 + 1
[8,7,5,6,4,2,3,1] => {{1,8},{2,7},{3,5},{4,6}}
=> {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> ? = 0 + 1
[8,7,6,4,5,2,3,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? = 0 + 1
[8,6,5,7,3,2,4,1] => {{1,8},{2,6},{3,5},{4,7}}
=> {{1,8},{2,5},{3,7},{4,6}}
=> {{1},{2,6},{3,5,7},{4,8}}
=> ? = 0 + 1
[6,7,8,5,2,3,4,1] => {{1,6},{2,7},{3,8},{4,5}}
=> {{1,6},{2,7},{3,8},{4,5}}
=> {{1,5},{2},{3},{4,6,7,8}}
=> ? = 1 + 1
[8,5,6,7,2,3,4,1] => {{1,8},{2,5},{3,6},{4,7}}
=> {{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2},{3,5,6,7},{4,8}}
=> ? = 0 + 1
[7,8,3,2,4,5,6,1] => {{1,7},{2,8},{3},{4},{5},{6}}
=> {{1,7},{2,8},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 0 + 1
[7,6,5,4,3,2,8,1] => {{1,7,8},{2,6},{3,5},{4}}
=> {{1,2,8},{3,7},{4,6},{5}}
=> {{1,2},{3,6},{4,7},{5,8}}
=> ? = 1 + 1
[6,5,7,4,3,2,8,1] => {{1,6},{2,5},{3,7,8},{4}}
=> {{1,2,6},{3,8},{4,7},{5}}
=> {{1,2},{3},{4,7},{5,6,8}}
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => {{1,6},{2,5,7,8},{3,4}}
=> {{1,2,4,7},{3,8},{5,6}}
=> ?
=> ? = 2 + 1
[5,4,6,3,7,2,8,1] => {{1,5,7,8},{2,4},{3,6}}
=> {{1,2,4,8},{3,6},{5,7}}
=> ?
=> ? = 1 + 1
[3,4,5,2,6,7,8,1] => {{1,3,5,6,7,8},{2,4}}
=> {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3 + 1
[5,2,3,4,6,7,8,1] => {{1,5,6,7,8},{2},{3},{4}}
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7,8}}
=> ? = 3 + 1
[3,4,2,5,6,7,8,1] => {{1,3},{2,4,5,6,7,8}}
=> {{1,2,3,4,5,7},{6,8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 4 + 1
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> {{1,2,3,4,5,8},{6},{7}}
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 4 + 1
[8,7,6,5,4,3,1,2] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 1 + 1
[7,8,6,5,4,3,1,2] => {{1,7},{2,8},{3,6},{4,5}}
=> {{1,7},{2,8},{3,6},{4,5}}
=> {{1,5},{2,6},{3},{4,7,8}}
=> ? = 1 + 1
[8,6,7,5,4,3,1,2] => {{1,8},{2,6},{3,7},{4,5}}
=> {{1,8},{2,6},{3,7},{4,5}}
=> {{1,5},{2},{3,6,7},{4,8}}
=> ? = 1 + 1
[8,7,5,6,4,3,1,2] => {{1,8},{2,7},{3,5},{4,6}}
=> {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> ? = 0 + 1
[8,7,6,4,5,3,1,2] => {{1,8},{2,7},{3,6},{4},{5}}
=> {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? = 0 + 1
[8,7,6,5,3,4,1,2] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 1 + 1
[8,7,5,6,3,4,1,2] => {{1,8},{2,7},{3,5},{4,6}}
=> {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> ? = 0 + 1
[7,8,5,6,3,4,1,2] => {{1,7},{2,8},{3,5},{4,6}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> ? = 0 + 1
[5,6,7,8,3,4,1,2] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 0 + 1
[7,8,3,4,5,6,1,2] => {{1,7},{2,8},{3},{4},{5},{6}}
=> {{1,7},{2,8},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 0 + 1
[8,7,6,5,4,2,1,3] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 1 + 1
[7,6,8,5,4,2,1,3] => {{1,7},{2,6},{3,8},{4,5}}
=> {{1,6},{2,8},{3,7},{4,5}}
=> {{1,5},{2},{3,7},{4,6,8}}
=> ? = 1 + 1
[8,7,6,5,4,1,2,3] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 1 + 1
[6,7,8,5,4,1,2,3] => {{1,6},{2,7},{3,8},{4,5}}
=> {{1,6},{2,7},{3,8},{4,5}}
=> {{1,5},{2},{3},{4,6,7,8}}
=> ? = 1 + 1
[8,5,6,7,4,1,2,3] => {{1,8},{2,5},{3,6},{4,7}}
=> {{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2},{3,5,6,7},{4,8}}
=> ? = 0 + 1
[8,7,6,4,5,1,2,3] => {{1,8},{2,7},{3,6},{4},{5}}
=> {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? = 0 + 1
[8,6,7,4,5,1,2,3] => {{1,8},{2,6},{3,7},{4},{5}}
=> {{1,8},{2,6},{3,7},{4},{5}}
=> ?
=> ? = 0 + 1
[8,7,4,5,6,1,2,3] => {{1,8},{2,7},{3,4,5,6}}
=> {{1,8},{2,7},{3,4,5,6}}
=> ?
=> ? = 3 + 1
[7,8,4,5,6,1,2,3] => {{1,7},{2,8},{3,4,5,6}}
=> {{1,7},{2,8},{3,4,5,6}}
=> ?
=> ? = 3 + 1
[6,7,4,5,8,1,2,3] => {{1,6},{2,7},{3,4,5,8}}
=> {{1,4,5,6},{2,7},{3,8}}
=> ?
=> ? = 2 + 1
[7,6,5,8,3,2,1,4] => {{1,7},{2,6},{3,5},{4,8}}
=> {{1,5},{2,8},{3,7},{4,6}}
=> {{1},{2,6},{3,7},{4,5,8}}
=> ? = 0 + 1
[7,5,6,8,2,3,1,4] => {{1,7},{2,5},{3,6},{4,8}}
=> {{1,5},{2,8},{3,6},{4,7}}
=> {{1},{2},{3,6,7},{4,5,8}}
=> ? = 0 + 1
Description
The cardinality of the first block of a set partition.
The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St001067
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 64%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[3,1,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,4,2,3] => [3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,4,3,2] => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[2,4,1,3] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[4,1,3,2] => [4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,1,3] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,4,3] => [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,3,2,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [5,2,4,3,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [3,4,2,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [4,5,2,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,2,1,3,8] => [4,7,3,5,2,6,1,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,5,6,4,3,2,1] => [7,6,5,3,4,1,2,8] => [5,4,3,7,2,6,1,8] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 0
[7,8,6,4,5,3,2,1] => [7,6,4,5,3,1,2,8] => [5,3,4,7,2,6,1,8] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 0
[8,6,7,4,5,3,2,1] => [7,6,4,5,2,3,1,8] => [6,3,4,5,2,7,1,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [7,6,4,3,2,1,5,8] => [4,3,7,5,2,6,1,8] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[6,5,7,4,8,3,2,1] => [7,6,4,2,1,3,5,8] => [6,3,4,2,7,5,1,8] => ?
=> ? = 0
[7,8,6,5,3,4,2,1] => [7,5,6,4,3,1,2,8] => [4,7,2,5,3,6,1,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,3,4,2,1] => [7,5,6,4,2,3,1,8] => [4,5,2,6,3,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1,8] => [6,2,5,4,3,7,1,8] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[6,7,8,3,4,5,2,1] => [7,4,5,6,1,2,3,8] => [6,2,4,7,3,5,1,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[7,8,6,5,4,2,3,1] => [6,7,5,4,3,1,2,8] => [4,5,3,6,1,7,2,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1,8] => [4,7,1,6,3,5,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,7,5,6,4,2,3,1] => [6,7,5,3,4,2,1,8] => [5,4,3,7,1,6,2,8] => ?
=> ? = 0
[8,7,6,4,5,2,3,1] => [6,7,4,5,3,2,1,8] => [5,3,4,7,1,6,2,8] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 0
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1,8] => [5,2,7,1,6,4,3,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0
[6,7,8,5,2,3,4,1] => [5,6,7,4,1,2,3,8] => [4,5,1,6,2,7,3,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1,8] => [6,4,2,7,1,5,3,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[7,8,3,2,4,5,6,1] => [4,3,5,6,7,1,2,8] => [6,1,4,7,2,3,5,8] => ?
=> ? = 0
[7,6,5,4,3,2,8,1] => [6,5,4,3,2,1,7,8] => [4,3,5,2,6,1,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1
[6,5,7,4,3,2,8,1] => [6,5,4,2,1,3,7,8] => [6,3,4,2,5,1,7,8] => ?
=> ? = 1
[6,5,4,3,7,2,8,1] => [6,4,3,2,1,5,7,8] => [3,4,2,6,5,1,7,8] => ?
=> ? = 2
[5,4,6,3,7,2,8,1] => [6,4,2,1,3,5,7,8] => [6,5,3,2,4,1,7,8] => ?
=> ? = 1
[3,4,5,2,6,7,8,1] => [4,1,2,3,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[5,2,3,4,6,7,8,1] => [2,3,4,1,5,6,7,8] => [4,1,2,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,3,2,5,6,7,8,1] => [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[3,4,2,5,6,7,8,1] => [3,1,2,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[4,2,3,5,6,7,8,1] => [2,3,1,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[3,2,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[8,7,6,5,4,3,1,2] => [8,6,5,4,3,2,1,7] => [4,5,3,6,2,8,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[7,8,6,5,4,3,1,2] => [8,6,5,4,3,1,2,7] => [4,5,3,8,7,2,6,1] => [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 1
[8,6,7,5,4,3,1,2] => [8,6,5,4,2,3,1,7] => [4,6,3,5,2,8,7,1] => [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[8,7,5,6,4,3,1,2] => [8,6,5,3,4,2,1,7] => [5,4,3,6,2,8,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 0
[8,7,6,4,5,3,1,2] => [8,6,4,5,3,2,1,7] => [5,3,4,6,2,8,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 0
[8,7,6,5,3,4,1,2] => [8,5,6,4,3,2,1,7] => [4,6,2,5,3,8,7,1] => [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[8,7,5,6,3,4,1,2] => [8,5,6,3,4,2,1,7] => [6,2,5,4,3,8,7,1] => ?
=> ? = 0
[7,8,5,6,3,4,1,2] => [8,5,6,3,4,1,2,7] => [8,7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [8,5,6,1,2,3,4,7] => [5,2,6,3,8,7,4,1] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [8,3,4,5,6,1,2,7] => [8,7,2,3,4,5,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,4,5,6,7,8,1,2] => [8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [6,8,5,4,3,2,1,7] => [4,5,3,8,7,1,6,2] => ?
=> ? = 1
[7,6,8,5,4,2,1,3] => [6,8,5,4,2,1,3,7] => [4,6,1,8,7,3,5,2] => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 1
[8,7,6,5,4,1,2,3] => [7,8,5,4,3,2,1,6] => [4,5,3,7,1,8,6,2] => ?
=> ? = 1
[6,7,8,5,4,1,2,3] => [7,8,5,4,1,2,3,6] => [4,7,3,5,1,8,6,2] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1
[8,5,6,7,4,1,2,3] => [7,8,5,2,3,4,1,6] => [5,3,7,1,8,6,4,2] => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 0
[8,7,6,4,5,1,2,3] => [7,8,4,5,3,2,1,6] => [5,3,4,7,1,8,6,2] => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> ? = 0
[8,6,7,4,5,1,2,3] => [7,8,4,5,2,3,1,6] => [7,1,8,6,3,4,5,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[8,7,4,5,6,1,2,3] => [7,8,3,4,5,2,1,6] => [3,4,5,7,1,8,6,2] => ?
=> ? = 3
[7,8,4,5,6,1,2,3] => [7,8,3,4,5,1,2,6] => [3,4,5,8,6,1,7,2] => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000441
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 91%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [3,2,1] => [1,2,3] => 2
[3,1,2] => [2,3,1] => [3,1,2] => [2,1,3] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => [4,1,2,3] => 2
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 3
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [3,1,2,4] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => [1,2,4,3] => 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => [2,1,3,4] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => [1,3,4,2] => 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 0
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 0
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => [2,3,1,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 3
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => [4,2,3,5,1] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => [2,3,5,4,1] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => [2,4,5,3,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => ? = 1
[1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [5,4,3,6,7,2,1] => ? = 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 1
[1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => [6,5,4,2,3,7,1] => ? = 1
[1,3,7,6,5,4,2] => [1,7,2,6,5,4,3] => [1,5,6,4,7,3,2] => [2,3,7,4,6,5,1] => ? = 1
[1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => [1,6,3,7,4,2,5] => [5,2,4,7,3,6,1] => ? = 0
[1,4,6,7,5,3,2] => [1,7,6,2,5,3,4] => [1,5,6,3,7,4,2] => [2,4,7,3,6,5,1] => ? = 0
[1,4,7,2,3,5,6] => [1,4,5,2,6,7,3] => [1,4,2,7,3,5,6] => [6,5,3,7,2,4,1] => ? = 0
[1,4,7,6,5,3,2] => [1,7,6,2,5,4,3] => [1,5,7,3,6,4,2] => [2,4,6,3,7,5,1] => ? = 0
[1,5,2,3,4,7,6] => [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [6,7,4,3,2,5,1] => ? = 1
[1,5,2,7,3,4,6] => [1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => [6,4,7,3,2,5,1] => ? = 0
[1,5,4,7,6,3,2] => [1,7,6,3,2,5,4] => [1,7,4,3,6,5,2] => [2,5,6,3,4,7,1] => ? = 2
[1,5,6,2,7,3,4] => [1,4,6,7,2,3,5] => [1,6,3,7,5,2,4] => [4,2,5,7,3,6,1] => ? = 0
[1,5,6,4,7,3,2] => [1,7,6,4,2,3,5] => [1,4,6,3,7,5,2] => [2,5,7,3,6,4,1] => ? = 0
[1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => [1,5,2,6,3,7,4] => [4,7,3,6,2,5,1] => ? = 0
[1,5,6,7,4,3,2] => [1,7,6,5,2,3,4] => [1,6,3,7,4,5,2] => [2,5,4,7,3,6,1] => ? = 0
[1,5,7,2,3,4,6] => [1,4,5,6,2,7,3] => [1,7,3,5,2,4,6] => [6,4,2,5,3,7,1] => ? = 0
[1,5,7,6,4,3,2] => [1,7,6,5,2,4,3] => [1,7,3,6,4,5,2] => [2,5,4,6,3,7,1] => ? = 0
[1,6,2,3,4,7,5] => [1,3,4,5,7,2,6] => [1,7,6,2,3,4,5] => [5,4,3,2,6,7,1] => ? = 1
[1,6,2,3,7,4,5] => [1,3,4,6,7,2,5] => [1,6,2,3,4,7,5] => [5,7,4,3,2,6,1] => ? = 0
[1,6,2,7,3,4,5] => [1,3,5,6,7,2,4] => [1,7,4,6,2,3,5] => [5,3,2,6,4,7,1] => ? = 0
[1,6,5,4,3,7,2] => [1,7,5,4,3,2,6] => [1,4,5,3,7,6,2] => [2,6,7,3,5,4,1] => ? = 1
[1,6,5,4,7,3,2] => [1,7,6,4,3,2,5] => [1,4,7,5,3,6,2] => [2,6,3,5,7,4,1] => ? = 0
[1,6,5,7,4,3,2] => [1,7,6,5,3,2,4] => [1,7,4,5,3,6,2] => [2,6,3,5,4,7,1] => ? = 0
[1,6,7,2,3,4,5] => [1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => [5,3,7,4,2,6,1] => ? = 0
[1,6,7,5,4,3,2] => [1,7,6,5,4,2,3] => [1,5,4,7,3,6,2] => [2,6,3,7,4,5,1] => ? = 1
[1,7,2,3,4,5,6] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => [6,5,4,3,2,7,1] => ? = 0
[1,7,3,5,6,4,2] => [1,7,3,6,4,5,2] => [1,3,6,5,4,7,2] => [2,7,4,5,6,3,1] => ? = 2
[1,7,3,6,5,4,2] => [1,7,3,6,5,4,2] => [1,3,5,6,4,7,2] => [2,7,4,6,5,3,1] => ? = 0
[1,7,4,3,6,5,2] => [1,7,4,3,6,5,2] => [1,4,3,6,5,7,2] => [2,7,5,6,3,4,1] => ? = 2
[1,7,4,5,3,6,2] => [1,7,5,3,4,6,2] => [1,5,4,3,6,7,2] => [2,7,6,3,4,5,1] => ? = 2
[1,7,4,5,6,3,2] => [1,7,6,3,4,5,2] => [1,6,5,4,3,7,2] => [2,7,3,4,5,6,1] => ? = 3
[1,7,4,6,5,3,2] => [1,7,6,3,5,4,2] => [1,5,6,4,3,7,2] => [2,7,3,4,6,5,1] => ? = 1
[1,7,5,4,3,6,2] => [1,7,5,4,3,6,2] => [1,4,5,3,6,7,2] => [2,7,6,3,5,4,1] => ? = 0
[1,7,5,4,6,3,2] => [1,7,6,4,3,5,2] => [1,4,6,5,3,7,2] => [2,7,3,5,6,4,1] => ? = 1
[1,7,6,4,5,3,2] => [1,7,6,4,5,3,2] => [1,4,5,6,3,7,2] => [2,7,3,6,5,4,1] => ? = 0
[1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => [2,7,3,6,4,5,1] => ? = 1
[2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => [2,1,3,4,7,5,6] => [6,5,7,4,3,1,2] => ? = 1
[2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => [2,1,3,7,4,5,6] => [6,5,4,7,3,1,2] => ? = 1
[2,1,6,3,4,5,7] => [2,1,4,5,6,3,7] => [2,1,6,3,4,5,7] => [7,5,4,3,6,1,2] => ? = 1
[2,1,7,3,4,5,6] => [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => [6,5,4,3,7,1,2] => ? = 1
[2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => [3,7,4,6,5,1,2] => ? = 1
[2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [4,3,2,1,5,7,6] => [6,7,5,1,2,3,4] => ? = 4
[2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [6,4,3,2,1,7,5] => [5,7,1,2,3,4,6] => ? = 3
[2,3,4,7,1,5,6] => [5,1,2,3,6,7,4] => [7,4,3,2,1,5,6] => [6,5,1,2,3,4,7] => ? = 3
[2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => [4,2,1,3,5,6,7] => [7,6,5,3,1,2,4] => ? = 1
[2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => [5,2,1,3,4,6,7] => [7,6,4,3,1,2,5] => ? = 1
[2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => [7,3,5,2,1,4,6] => [6,4,1,2,5,3,7] => ? = 1
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St001640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 64%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,3] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 2
[3,1,2] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,3,2,4] => [2,4,3,1] => [3,4,2,1] => [4,2,1,3] => 1
[1,3,4,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,4,2,3] => [2,4,1,3] => [4,2,1,3] => [3,4,2,1] => 0
[1,4,3,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,3,4] => [3,2,4,1] => [2,4,3,1] => [4,3,1,2] => 1
[2,1,4,3] => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[2,3,1,4] => [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[2,4,1,3] => [4,2,1,3] => [2,4,1,3] => [3,4,1,2] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [3,4,2,1] => [4,2,3,1] => [3,1,4,2] => 0
[3,1,4,2] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[3,2,1,4] => [4,3,2,1] => [3,2,4,1] => [4,1,3,2] => 0
[3,2,4,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[3,4,2,1] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[4,1,2,3] => [3,4,1,2] => [4,1,3,2] => [3,2,4,1] => 0
[4,1,3,2] => [3,1,4,2] => [4,3,1,2] => [2,4,3,1] => 0
[4,2,1,3] => [4,3,1,2] => [3,1,4,2] => [4,2,3,1] => 0
[4,2,3,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[4,3,1,2] => [4,1,3,2] => [3,4,1,2] => [2,4,1,3] => 1
[4,3,2,1] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => [5,3,2,1,4] => 1
[1,2,4,5,3] => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
[1,2,5,3,4] => [2,3,5,1,4] => [5,3,2,1,4] => [4,5,3,2,1] => 0
[1,2,5,4,3] => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => [5,4,2,1,3] => 1
[1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => [4,2,1,3,5] => 2
[1,3,4,2,5] => [2,5,3,4,1] => [3,4,5,2,1] => [5,2,1,3,4] => 2
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
[1,3,5,2,4] => [2,5,3,1,4] => [3,5,2,1,4] => [4,5,2,1,3] => 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,4,2,3,5] => [2,4,5,3,1] => [5,3,4,2,1] => [4,2,1,5,3] => 0
[1,4,2,5,3] => [2,4,1,3,5] => [4,2,1,3,5] => [3,4,2,1,5] => 1
[1,4,3,2,5] => [2,5,4,3,1] => [4,3,5,2,1] => [5,2,1,4,3] => 0
[1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,4,5,2,3] => [2,5,1,3,4] => [5,2,1,3,4] => [3,4,5,2,1] => 0
[1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [2,3,4,5,6,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 1
[1,2,3,4,6,5,7] => [2,3,4,5,7,6,1] => [6,7,5,4,3,2,1] => [7,5,4,3,2,1,6] => ? = 1
[1,2,3,6,4,7,5] => [2,3,4,6,1,5,7] => [6,4,3,2,1,5,7] => [5,6,4,3,2,1,7] => ? = 1
[1,2,6,3,4,7,5] => [2,3,5,6,1,4,7] => [6,3,2,1,5,4,7] => [5,4,6,3,2,1,7] => ? = 1
[1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => [3,7,6,5,4,2,1] => [7,6,5,4,2,1,3] => ? = 1
[1,3,7,2,4,5,6] => [2,5,3,6,7,1,4] => [3,7,2,1,6,5,4] => [6,5,4,7,2,1,3] => ? = 1
[1,3,7,6,5,4,2] => [2,1,3,7,6,5,4] => [2,1,3,6,5,7,4] => [2,1,3,7,4,6,5] => ? = 1
[1,4,6,7,2,3,5] => [2,6,7,3,1,4,5] => [7,3,6,2,1,4,5] => [4,5,6,2,1,7,3] => ? = 0
[1,4,6,7,5,3,2] => [2,1,7,3,6,4,5] => [2,1,6,3,4,7,5] => [2,1,4,7,5,6,3] => ? = 0
[1,4,7,2,3,5,6] => [2,5,6,3,7,1,4] => [7,3,6,5,2,1,4] => [4,6,5,2,1,7,3] => ? = 0
[1,4,7,6,5,3,2] => [2,1,7,3,6,5,4] => [2,1,6,5,7,3,4] => [2,1,4,7,3,6,5] => ? = 0
[1,5,2,3,4,7,6] => [2,4,5,6,3,1,7] => [6,3,5,4,2,1,7] => [5,4,2,1,6,3,7] => ? = 1
[1,5,2,7,3,4,6] => [2,4,6,7,3,1,5] => [7,3,6,4,2,1,5] => [5,6,4,2,1,7,3] => ? = 0
[1,5,4,7,6,3,2] => [2,1,7,4,3,6,5] => [2,1,4,6,7,3,5] => [2,1,5,7,3,4,6] => ? = 2
[1,5,6,2,7,3,4] => [2,5,7,1,3,4,6] => [7,2,1,5,3,4,6] => [4,5,3,6,7,2,1] => ? = 0
[1,5,6,4,7,3,2] => [2,1,7,5,3,4,6] => [2,1,5,3,7,4,6] => [2,1,6,7,4,5,3] => ? = 0
[1,5,6,7,2,3,4] => [2,6,7,1,3,4,5] => [7,2,1,6,3,4,5] => [4,5,6,3,7,2,1] => ? = 0
[1,5,6,7,4,3,2] => [2,1,7,6,3,4,5] => [2,1,6,3,7,4,5] => [2,1,5,7,4,6,3] => ? = 0
[1,5,7,2,3,4,6] => [2,5,6,7,3,1,4] => [7,3,6,2,1,5,4] => [5,4,6,2,1,7,3] => ? = 0
[1,5,7,6,4,3,2] => [2,1,7,6,3,5,4] => [2,1,6,3,5,7,4] => [2,1,7,4,5,6,3] => ? = 0
[1,6,2,3,4,7,5] => [2,4,5,6,1,3,7] => [6,2,1,5,4,3,7] => [5,4,3,6,2,1,7] => ? = 1
[1,6,2,3,7,4,5] => [2,4,5,7,1,3,6] => [7,2,1,5,4,3,6] => [5,4,3,6,7,2,1] => ? = 0
[1,6,2,7,3,4,5] => [2,4,6,7,1,3,5] => [7,2,1,6,4,3,5] => [5,6,4,3,7,2,1] => ? = 0
[1,6,5,4,3,7,2] => [2,1,6,5,4,3,7] => [2,1,5,4,6,3,7] => [2,1,6,3,5,4,7] => ? = 1
[1,6,5,4,7,3,2] => [2,1,7,5,4,3,6] => [2,1,5,4,7,3,6] => [2,1,6,7,3,5,4] => ? = 0
[1,6,5,7,4,3,2] => [2,1,7,6,4,3,5] => [2,1,6,4,7,3,5] => [2,1,5,7,3,6,4] => ? = 0
[1,6,7,2,3,4,5] => [2,5,6,7,1,3,4] => [7,2,1,6,3,5,4] => [5,4,6,3,7,2,1] => ? = 0
[1,6,7,5,4,3,2] => [2,1,7,6,5,3,4] => [2,1,5,6,3,7,4] => [2,1,7,4,6,3,5] => ? = 1
[1,7,2,3,4,5,6] => [2,4,5,6,7,1,3] => [7,2,1,6,5,4,3] => [6,5,4,3,7,2,1] => ? = 0
[1,7,3,5,6,4,2] => [2,1,4,7,5,6,3] => [2,1,5,6,7,4,3] => [2,1,7,4,3,5,6] => ? = 2
[1,7,3,6,5,4,2] => [2,1,4,7,6,5,3] => [2,1,6,5,7,4,3] => [2,1,7,4,3,6,5] => ? = 0
[1,7,4,3,6,5,2] => [2,1,5,4,7,6,3] => [2,1,4,6,7,5,3] => [2,1,7,5,3,4,6] => ? = 2
[1,7,4,5,3,6,2] => [2,1,6,4,5,7,3] => [2,1,4,5,7,6,3] => [2,1,7,6,3,4,5] => ? = 2
[1,7,4,5,6,3,2] => [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => ? = 3
[1,7,4,6,5,3,2] => [2,1,7,4,6,5,3] => [2,1,4,6,5,7,3] => [2,1,7,3,4,6,5] => ? = 1
[1,7,5,4,3,6,2] => [2,1,6,5,4,7,3] => [2,1,5,4,7,6,3] => [2,1,7,6,3,5,4] => ? = 0
[1,7,5,4,6,3,2] => [2,1,7,5,4,6,3] => [2,1,5,4,6,7,3] => [2,1,7,3,5,4,6] => ? = 1
[1,7,6,4,5,3,2] => [2,1,7,5,6,4,3] => [2,1,6,5,4,7,3] => [2,1,7,3,6,5,4] => ? = 0
[1,7,6,5,4,3,2] => [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => [2,1,7,3,6,4,5] => ? = 1
[2,1,3,4,5,6,7] => [3,2,4,5,6,7,1] => [2,7,6,5,4,3,1] => [7,6,5,4,3,1,2] => ? = 1
[2,1,3,4,7,5,6] => [3,2,4,5,7,1,6] => [2,7,5,4,3,1,6] => [6,7,5,4,3,1,2] => ? = 1
[2,1,3,7,4,5,6] => [3,2,4,6,7,1,5] => [2,7,4,3,1,6,5] => [6,5,7,4,3,1,2] => ? = 1
[2,1,6,3,4,5,7] => [3,2,5,6,7,4,1] => [2,7,4,6,5,3,1] => [6,5,3,1,2,7,4] => ? = 1
[2,1,7,3,4,5,6] => [3,2,5,6,7,1,4] => [2,7,3,1,6,5,4] => [6,5,4,7,3,1,2] => ? = 1
[2,1,7,6,5,4,3] => [3,2,1,7,6,5,4] => [2,3,1,6,5,7,4] => [3,1,2,7,4,6,5] => ? = 1
[2,3,1,4,5,6,7] => [4,2,3,5,6,7,1] => [2,3,7,6,5,4,1] => [7,6,5,4,1,2,3] => ? = 2
[2,3,4,1,5,6,7] => [5,2,3,4,6,7,1] => [2,3,4,7,6,5,1] => [7,6,5,1,2,3,4] => ? = 3
[2,3,4,1,5,7,6] => [5,2,3,4,6,1,7] => [2,3,4,6,5,1,7] => [6,5,1,2,3,4,7] => ? = 4
[2,3,4,5,1,6,7] => [6,2,3,4,5,7,1] => [2,3,4,5,7,6,1] => [7,6,1,2,3,4,5] => ? = 4
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St001223
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 55%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [2,1] => [2,1] => [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[3,1,2] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,4,2,3] => [3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,4,3,2] => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[2,4,1,3] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[4,1,3,2] => [4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,1,3] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,4,3] => [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,3,2,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [5,2,4,3,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [3,4,2,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [4,5,2,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [6,7,1,2,3,5,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [5,3,6,7,1,2,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => [2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,7,2,4,5,6] => [4,2,5,6,7,3,1] => [2,7,1,4,6,3,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,7,6,5,4,2] => [7,2,6,5,4,3,1] => [2,5,4,6,3,7,1] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,4,6,7,2,3,5] => [5,6,2,7,3,4,1] => [7,1,5,3,2,6,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,4,6,7,5,3,2] => [7,6,2,5,3,4,1] => [6,4,5,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
[1,4,7,2,3,5,6] => [4,5,2,6,7,3,1] => [7,1,4,6,3,2,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,4,7,6,5,3,2] => [7,6,2,5,4,3,1] => [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,2,3,4,7,6] => [3,4,5,2,7,6,1] => [4,2,6,7,1,3,5] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,5,2,7,3,4,6] => [3,5,6,2,7,4,1] => [7,1,3,6,4,2,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,5,4,7,6,3,2] => [7,6,3,2,5,4,1] => [3,5,6,4,2,7,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,5,6,2,7,3,4] => [4,6,7,2,3,5,1] => [7,1,4,2,6,5,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,5,6,4,7,3,2] => [7,6,4,2,3,5,1] => [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
[1,5,6,7,2,3,4] => [5,6,7,2,3,4,1] => [6,4,2,7,1,5,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 0
[1,5,6,7,4,3,2] => [7,6,5,2,3,4,1] => [5,3,6,4,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,7,2,3,4,6] => [4,5,6,2,7,3,1] => [6,3,7,1,4,2,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0
[1,5,7,6,4,3,2] => [7,6,5,2,4,3,1] => [6,3,5,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
[1,6,2,3,4,7,5] => [3,4,5,7,2,6,1] => [6,7,1,3,5,2,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,6,2,3,7,4,5] => [3,4,6,7,2,5,1] => [7,1,3,6,5,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,6,2,7,3,4,5] => [3,5,6,7,2,4,1] => [5,2,7,1,3,6,4] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0
[1,6,5,4,3,7,2] => [7,5,4,3,2,6,1] => [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,5,4,7,3,2] => [7,6,4,3,2,5,1] => [4,3,6,5,2,7,1] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,6,5,7,4,3,2] => [7,6,5,3,2,4,1] => [6,4,3,5,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
[1,6,7,2,3,4,5] => [4,5,6,7,2,3,1] => [5,2,6,3,7,1,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0
[1,6,7,5,4,3,2] => [7,6,5,4,2,3,1] => [4,6,3,5,2,7,1] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,7,2,3,4,5,6] => [3,4,5,6,7,2,1] => [6,2,4,7,1,3,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 0
[1,7,3,5,6,4,2] => [7,3,6,4,5,2,1] => [4,5,6,2,3,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,7,3,6,5,4,2] => [7,3,6,5,4,2,1] => [5,4,6,2,3,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 0
[1,7,4,3,6,5,2] => [7,4,3,6,5,2,1] => [3,5,6,2,4,7,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,7,4,5,3,6,2] => [7,5,3,4,6,2,1] => [3,4,6,2,5,7,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,7,4,5,6,3,2] => [7,6,3,4,5,2,1] => [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,7,4,6,5,3,2] => [7,6,3,5,4,2,1] => [3,5,4,6,2,7,1] => [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,7,5,4,3,6,2] => [7,5,4,3,6,2,1] => [4,3,6,2,5,7,1] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,7,5,4,6,3,2] => [7,6,4,3,5,2,1] => [4,3,5,6,2,7,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,7,6,4,5,3,2] => [7,6,4,5,3,2,1] => [5,3,4,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 0
[1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,3,4,7,5,6] => [1,3,4,6,7,5,2] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,3,7,4,5,6] => [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[2,1,6,3,4,5,7] => [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[2,1,7,3,4,5,6] => [1,4,5,6,7,3,2] => [1,7,2,4,6,3,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,7,6,5,4,3] => [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[2,3,1,4,5,6,7] => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,4,1,5,6,7] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[2,3,4,1,5,7,6] => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
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St000475The number of parts equal to 1 in a partition.
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