Your data matches 170 different statistics following compositions of up to 3 maps.
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Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000586: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000609: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000039
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000039: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000065
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
Mp00137: Dyck paths to symmetric ASMAlternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
Description
The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St000355
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000355: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000359
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St000359: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000369
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
Mp00121: Dyck paths Cori-Le Borgne involutionDyck paths
St000369: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
Description
The dinv deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$ In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$. See also [[St000376]] for the bounce deficit.
Matching statistic: St000491
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
Mp00221: Set partitions conjugateSet partitions
St000491: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1,2,3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,3,4,5},{2}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,4,5},{2,3}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,5},{2,3,4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2,3,4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,5},{2,4},{3}}
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,5},{2,3},{4}}
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> {{1},{2,5},{3,4}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1},{2,5},{3},{4}}
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
Description
The number of inversions of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000496
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
Mp00221: Set partitions conjugateSet partitions
St000496: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1,2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> {{1},{2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1,2,3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,3,4,5},{2}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,4,5},{2,3}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,5},{2,3,4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2,3,4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,5},{2,4},{3}}
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,5},{2,3},{4}}
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> {{1},{2,5},{3,4}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1},{2,5},{3},{4}}
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
Description
The rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
Mp00174: Set partitions dual major index to intertwining numberSet partitions
St000572: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3},{2,4}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3,4},{2}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4},{2,5},{3}}
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2},{3,5}}
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,5},{2},{4}}
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> {{1},{2,4,5},{3}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,5},{2,4}}
=> 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,3},{2,4,5}}
=> 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
Description
The dimension exponent of a set partition. This is $$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$ where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$. It is thus equal to the difference [[St000728]] - [[St000211]]. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
The following 160 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001822The number of alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000068The number of minimal elements in a poset. St000090The variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000254The nesting number of a set partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001488The number of corners of a skew partition. St001623The number of doubly irreducible elements of a lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000528The height of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001061The number of indices that are both descents and recoils of a permutation. St001470The cyclic holeyness of a permutation. St001850The number of Hecke atoms of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000234The number of global ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000729The minimal arc length of a set partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001712The number of natural descents of a standard Young tableau. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000222The number of alignments in the permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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 $eA$. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001535The number of cyclic alignments of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001841The number of inversions of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001911A descent variant minus the number of inversions. St001928The number of non-overlapping descents in a permutation. St000080The rank of the poset. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000570The Edelman-Greene number of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001424The number of distinct squares in a binary word. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St000519The largest length of a factor maximising the subword complexity. St000677The standardized bi-alternating inversion number of a permutation. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001375The pancake length of a permutation. St001377The major index minus the number of inversions of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001705The number of occurrences of the pattern 2413 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000304The load of a permutation. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001077The prefix exchange distance of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001718The number of non-empty open intervals in a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001848The atomic length of a signed permutation. St000728The dimension of a set partition. St000008The major index of the composition. St000154The sum of the descent bottoms of a permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000305The inverse major index of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001671Haglund's hag of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000391The sum of the positions of the ones in a binary word. St001684The reduced word complexity of a permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.