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Matching statistic: St000864
St000864: 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] => 0
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
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: St000340
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000698
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%
Values
[1] => [] => []
=> []
=> ? = 0
[1,2] => [1] => [1,0]
=> []
=> ? ∊ {0,1}
[2,1] => [1] => [1,0]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,2}
[1,3,2] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,2}
[2,1,3] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2}
[2,3,1] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2}
[3,1,2] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,2}
[3,2,1] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2}
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,2,2,2,3}
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,5,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,5,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,5,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,5,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,1,5,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,3,5,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,4,1,3,5] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,1,5,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,1,5] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,5,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,5,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,5,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,2,3,5] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,2,5,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,3,2,5] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,3,5,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,5,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,5,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,1,3,5] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,1,5,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,3,1,5] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,3,5,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,5,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,5,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,1,2,5] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,1,5,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,2,5,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,5,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,5,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,5,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,5,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,5,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000708
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,3,1] => [3]
=> []
=> ? ∊ {0,0,1,1,2}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0,1,1,2}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,3,2,1] => [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> 2
[2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> 2
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> 2
[3,4,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,3,1] => [3]
=> []
=> ? ∊ {0,0,1,1,2}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0,1,1,2}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,3,2,1] => [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> 2
[2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> 2
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> 2
[3,4,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000668
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,4,1,2] => [2,2]
=> [2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,3,2,1] => [2,2]
=> [2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[2,1,4,5,3] => [3,2]
=> [2]
=> [2]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [2]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [2]
=> 2
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> [2]
=> 2
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> [2]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [2]
=> 2
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [3]
=> 3
[3,4,1,5,2] => [3,2]
=> [2]
=> [2]
=> 2
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> [2]
=> 2
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000681
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[3,4,1,2] => [2,2]
=> [2]
=> [2]
=> 1
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,3}
[4,3,2,1] => [2,2]
=> [2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [2]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [2]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [2]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> [2]
=> 1
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> [2]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [2]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [3]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [2]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> [2]
=> 1
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000939
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 2
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 2
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,1,1]
=> 3
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 2
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4}
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000460
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [[1],[]]
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[2,1] => [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,2}
[1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,2}
[2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,1,1,2}
[3,1,2] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,1,2}
[3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,1,2}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 3
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 3
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000771
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [2] => ([],2)
=> ? = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2} + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,2} + 1
[3,2,1] => [1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {1,1,2} + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,2,2,2,3} + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000259The diameter of a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000937The number of positive values of the symmetric group character corresponding to the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000815The number of semistandard Young tableaux of partition weight of given shape. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000366The number of double descents of a permutation. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St001875The number of simple modules with projective dimension at most 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001863The number of weak excedances of a signed permutation. St001948The number of augmented double ascents of a permutation. St001621The number of atoms of a lattice.
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