Your data matches 244 different statistics following compositions of up to 3 maps.
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Matching statistic: St000058
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
Mp00080: Set partitions to permutationPermutations
St000058: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> {{1}}
=> [1] => 1
[2]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 2
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 3
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 2
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 4
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 3
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 4
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 6
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 4
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => 6
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 1
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
St000668: Integer partitions ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1]
=> ? = 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 6
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 4
[4,1,1]
=> 4
[3,3]
=> 3
[3,2,1]
=> 6
[3,1,1,1]
=> 3
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000306
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000306: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[3]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[4]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[3,1]
=> 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[2,2]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,1,1]
=> 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[5]
=> 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[4,1]
=> 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[3,2]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1]
=> 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 6
[2,2,1]
=> 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[6]
=> 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[5,1]
=> 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[4,2]
=> 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {3,4,6}
[3,3]
=> 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[3,2,1]
=> 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {3,4,6}
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,4,6}
[2,2,2]
=> 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001330: Graphs ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 2
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [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)
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,6}
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,6}
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,6}
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001232: Dyck paths ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> ? = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ? = 6 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,4,6} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,4,6} - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,4,6} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000184
Mp00202: Integer partitions first row removalInteger partitions
Mp00323: Integer partitions Loehr-Warrington inverseInteger partitions
St000184: Integer partitions ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ? = 2
[1,1]
=> [1]
=> [1]
=> 1
[3]
=> []
=> ?
=> ? = 3
[2,1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> 2
[4]
=> []
=> ?
=> ? = 4
[3,1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[5]
=> []
=> ?
=> ? = 5
[4,1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [1,1]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> 2
[2,2,1]
=> [2,1]
=> [1,1,1]
=> 6
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
[6]
=> []
=> ?
=> ? = 6
[5,1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [1,1]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> 2
[3,3]
=> [3]
=> [2,1]
=> 2
[3,2,1]
=> [2,1]
=> [1,1,1]
=> 6
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[2,2,2]
=> [2,2]
=> [2,1,1]
=> 4
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 5
Description
The size of the centralizer of any permutation of given cycle type. The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$: $$C_g = \{h \in G : hgh^{-1} = g\}.$$ Its size thus depends only on the conjugacy class of $g$. The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is $$|C| = \Pi j^{a_j} a_j!$$ For example, for any permutation with cycle type $\lambda = (3,2,2,1)$, $$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$ There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.
Matching statistic: St001291
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St001291: Dyck paths ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 2
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[4]
=> []
=> []
=> []
=> ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[5]
=> []
=> []
=> []
=> ? = 6
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[6]
=> []
=> []
=> []
=> ? = 6
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St001497
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St001497: Permutations ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> [] => ? = 1
[2]
=> []
=> []
=> [] => ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => ? = 2
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4
[5]
=> []
=> []
=> [] => ? = 6
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[6]
=> []
=> []
=> [] => ? = 6
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 3
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 6
Description
The position of the largest weak excedence of a permutation.
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000454: Graphs ⟶ ℤResult quality: 72% values known / values provided: 72%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,6} - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,6} - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [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 = 5 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {2,2,3,4,6} - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,4,6} - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,4,6} - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,4,6} - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,4,6} - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001637
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St001637: Posets ⟶ ℤResult quality: 67% values known / values provided: 69%distinct values known / distinct values provided: 67%
Values
[1]
=> 1 => ([(0,1)],2)
=> 1
[2]
=> 0 => ([(0,1)],2)
=> 1
[1,1]
=> 11 => ([(0,2),(2,1)],3)
=> 2
[3]
=> 1 => ([(0,1)],2)
=> 1
[2,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1]
=> 111 => ([(0,3),(2,1),(3,2)],4)
=> 3
[4]
=> 0 => ([(0,1)],2)
=> 1
[3,1]
=> 11 => ([(0,2),(2,1)],3)
=> 2
[2,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2
[2,1,1]
=> 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 3
[1,1,1,1]
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[5]
=> 1 => ([(0,1)],2)
=> 1
[4,1]
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1]
=> 111 => ([(0,3),(2,1),(3,2)],4)
=> 3
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? ∊ {4,5,6}
[2,1,1,1]
=> 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,5,6}
[1,1,1,1,1]
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {4,5,6}
[6]
=> 0 => ([(0,1)],2)
=> 1
[5,1]
=> 11 => ([(0,2),(2,1)],3)
=> 2
[4,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2
[4,1,1]
=> 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? ∊ {3,4,5,6,6}
[3,3]
=> 11 => ([(0,2),(2,1)],3)
=> 2
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? ∊ {3,4,5,6,6}
[3,1,1,1]
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3
[2,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {3,4,5,6,6}
[2,1,1,1,1]
=> 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {3,4,5,6,6}
[1,1,1,1,1,1]
=> 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {3,4,5,6,6}
Description
The number of (upper) dissectors of a poset.
The following 234 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001668The number of points of the poset minus the width of the poset. St000456The monochromatic index of a connected graph. St001638The book thickness of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001644The dimension of a graph. St001959The product of the heights of the peaks of a Dyck path. St000392The length of the longest run of ones in a binary word. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000628The balance of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000982The length of the longest constant subword. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St001937The size of the center of a parking function. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001437The flex of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001712The number of natural descents of a standard Young tableau. St001948The number of augmented double ascents of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St001060The distinguishing index of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a 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. St001814The number of partitions interlacing the given partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000006The dinv of a Dyck path. St000766The number of inversions of an integer composition. St000942The number of critical left to right maxima of the parking functions. 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 $eA$. St001863The number of weak excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001423The number of distinct cubes in a binary word. St001424The number of distinct squares in a binary word. St001488The number of corners of a skew partition. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000735The last entry on the main diagonal of a standard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001896The number of right descents of a signed permutations. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001946The number of descents in a parking function. St000031The number of cycles in the cycle decomposition of a permutation. St000141The maximum drop size of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001096The size of the overlap set of a permutation. 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 $eA$. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000662The staircase size of the code of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000438The position of the last up step in a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000675The number of centered multitunnels of a Dyck path. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St000981The length of the longest zigzag subpath. St001118The acyclic chromatic index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000056The decomposition (or block) number of a permutation. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001128The exponens consonantiae of a partition. St001151The number of blocks with odd minimum. St001461The number of topologically connected components of the chord diagram of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000173The segment statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001727The number of invisible inversions of a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001843The Z-index of a set partition. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000834The number of right outer peaks of a permutation. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001568The smallest positive integer that does not appear twice in the partition. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001487The number of inner corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000562The number of internal points of 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. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000779The tier of a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001469The holeyness of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001935The number of ascents in a parking function. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000365The number of double ascents of a permutation. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 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. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000230Sum of the minimal elements of the blocks of a set partition. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000307The number of rowmotion orbits of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001811The Castelnuovo-Mumford regularity of a permutation. St000717The number of ordinal summands of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral.