Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000864
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 4
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 3
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 3
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 3
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
Description
The number of circled entries of the shifted recording tableau of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of circled entries in $Q$.
Matching statistic: St000359
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [2,1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 3
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 4
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 3
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 3
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 3
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 2
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 2
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 3
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 2
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 2
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 1
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,3,5,2,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,3,5,2,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,3,5,4,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,3,6,2,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,3,6,2,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,3,6,4,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,3,6,4,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,3,6,5,2,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 2
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 3
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 2
[1,4,2,5,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1
[1,4,2,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 2
[1,4,2,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1
[1,4,2,6,5,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 1
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 2
[1,4,3,2,6,5] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 1
[1,4,3,5,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1
[1,4,3,5,6,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 2
[1,4,3,6,2,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 1
[1,4,3,6,5,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 1
[1,4,5,2,3,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,4,5,2,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,4,5,3,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,4,5,3,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,4,6,2,3,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 3
[1,4,6,2,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
[1,4,6,3,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 2
[1,4,6,3,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 2
Description
The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St001223
Mapping
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
 => [1,0]
 => [1,1,0,0]
 => 0
[1,2] => [.,[.,.]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [[.,.],.]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
[4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 4
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 3
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 4
[1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 3
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 2
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 4
[1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 3
[1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3
[1,2,5,3,6,4] => [.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 2
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 2
[1,2,5,4,6,3] => [.,[.,[[[.,.],[.,.]],.]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 2
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,2,5,6,4,3] => [.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,2,6,3,4,5] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3
[1,2,6,3,5,4] => [.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 2
[1,2,6,4,3,5] => [.,[.,[[[.,.],.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 2
[1,2,6,4,5,3] => [.,[.,[[[.,.],[.,.]],.]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 2
[1,2,6,5,3,4] => [.,[.,[[[.,.],.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 2
[1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 3
[1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,3,2,5,6,4] => [.,[[.,.],[[.,[.,.]],.]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
[1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,3,4,2,6,5] => [.,[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
[1,3,5,2,4,6] => [.,[[.,[.,.]],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,3,5,2,6,4] => [.,[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,3,5,4,6,2] => [.,[[.,[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,3,5,6,2,4] => [.,[[.,[.,[.,.]]],[.,.]]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 3
[1,3,6,2,4,5] => [.,[[.,[.,.]],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,3,6,2,5,4] => [.,[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[1,3,6,4,2,5] => [.,[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,3,6,4,5,2] => [.,[[.,[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,3,6,5,2,4] => [.,[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
[1,4,2,3,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 3
[1,4,2,3,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 2
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.
Matching statistic: St001233
(load all 2 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 3 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 5 + 1
[1,2,3,4,6,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 4 + 1
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,2,3,5,6,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 4 + 1
[1,2,3,6,4,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,2,3,6,5,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,4,5,3,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,2,4,5,6,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 4 + 1
[1,2,4,6,3,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,2,4,6,5,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,5,6,3,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,2,5,6,4,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 1 + 1
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 1 + 1
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => ? = 1 + 1
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,3,4,5,2,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,3,4,5,6,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 4 + 1
[1,3,4,6,2,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,3,4,6,5,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,3,5,2,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,3,5,2,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,3,5,4,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,3,5,6,2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,3,5,6,4,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,3,6,2,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,3,6,2,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,3,6,4,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,3,6,4,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 2 + 1
[1,3,6,5,2,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 2 + 1
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
Matching statistic: St001948
(load all 19 compositions to match this statistic)
Mapping
St001948: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 83%
Values
[1] => ? = 0
[1,2] => 1
[2,1] => 0
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 3
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
[1,2,3,4,5,6] => ? = 5
[1,2,3,4,6,5] => ? = 4
[1,2,3,5,4,6] => ? = 3
[1,2,3,5,6,4] => ? = 4
[1,2,3,6,4,5] => ? = 3
[1,2,3,6,5,4] => ? = 3
[1,2,4,3,5,6] => ? = 3
[1,2,4,3,6,5] => ? = 2
[1,2,4,5,3,6] => ? = 3
[1,2,4,5,6,3] => ? = 4
[1,2,4,6,3,5] => ? = 3
[1,2,4,6,5,3] => ? = 3
[1,2,5,3,4,6] => ? = 3
[1,2,5,3,6,4] => ? = 2
[1,2,5,4,3,6] => ? = 2
[1,2,5,4,6,3] => ? = 2
[1,2,5,6,3,4] => ? = 3
[1,2,5,6,4,3] => ? = 3
[1,2,6,3,4,5] => ? = 3
[1,2,6,3,5,4] => ? = 2
[1,2,6,4,3,5] => ? = 2
[1,2,6,4,5,3] => ? = 2
[1,2,6,5,3,4] => ? = 2
[1,2,6,5,4,3] => ? = 2
[1,3,2,4,5,6] => ? = 3
[1,3,2,4,6,5] => ? = 2
[1,3,2,5,4,6] => ? = 1
[1,3,2,5,6,4] => ? = 2
[1,3,2,6,4,5] => ? = 1
[1,3,2,6,5,4] => ? = 1
[1,3,4,2,5,6] => ? = 3
[1,3,4,2,6,5] => ? = 2
[1,3,4,5,2,6] => ? = 3
[1,3,4,5,6,2] => ? = 4
[1,3,4,6,2,5] => ? = 3
[1,3,4,6,5,2] => ? = 3
[1,3,5,2,4,6] => ? = 3
[1,3,5,2,6,4] => ? = 2
[1,3,5,4,2,6] => ? = 2
[1,3,5,4,6,2] => ? = 2
[1,3,5,6,2,4] => ? = 3
[1,3,5,6,4,2] => ? = 3
[1,3,6,2,4,5] => ? = 3
[1,3,6,2,5,4] => ? = 2
[1,3,6,4,2,5] => ? = 2
[1,3,6,4,5,2] => ? = 2
[1,3,6,5,2,4] => ? = 2
[1,3,6,5,4,2] => ? = 2
[1,4,2,3,5,6] => ? = 3
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.