Your data matches 126 different statistics following compositions of up to 3 maps.
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Mp00080: Set partitions to permutationPermutations
St000002: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => 1 = 2 - 1
{{1,2,3,4}}
=> [2,3,4,1] => 1 = 2 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 4 = 5 - 1
Description
The number of occurrences of the pattern 123 in a permutation.
Mp00080: Set partitions to permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000255: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 5
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 1
Description
The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
Mp00128: Set partitions to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000095: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> 0 = 1 - 1
{{1,2}}
=> [2] => ([],2)
=> 0 = 1 - 1
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
{{1,2,3}}
=> [3] => ([],3)
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1,2,3,4}}
=> [4] => ([],4)
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 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
Description
The number of triangles of a graph. A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Mp00080: Set partitions to permutationPermutations
Mp00069: Permutations complementPermutations
St000119: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [1,2] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [2,1] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [2,1,3] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,3,1] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => 1 = 2 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1,2,4] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,3,1,4] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
Description
The number of occurrences of the pattern 321 in a permutation.
Mp00080: Set partitions to permutationPermutations
Mp00277: Permutations catalanizationPermutations
St000430: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 1 = 2 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [3,4,2,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
Description
The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.
Mp00128: Set partitions to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001317: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> 0 = 1 - 1
{{1,2}}
=> [2] => ([],2)
=> 0 = 1 - 1
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
{{1,2,3}}
=> [3] => ([],3)
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1,2,3,4}}
=> [4] => ([],4)
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 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
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001319
Mp00128: Set partitions to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001319: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> 0 = 1 - 1
{{1,2}}
=> [2] => ([],2)
=> 0 = 1 - 1
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
{{1,2,3}}
=> [3] => ([],3)
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1,2,3,4}}
=> [4] => ([],4)
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 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
Description
The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. A graph is a disjoint union of isolated vertices and a star if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Mp00128: Set partitions to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001328: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> 0 = 1 - 1
{{1,2}}
=> [2] => ([],2)
=> 0 = 1 - 1
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
{{1,2,3}}
=> [3] => ([],3)
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1,2,3,4}}
=> [4] => ([],4)
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 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
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Mp00080: Set partitions to permutationPermutations
Mp00126: Permutations cactus evacuationPermutations
St001411: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [2,1,3] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,3,1] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,1,3,4] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,1,3] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,3,4,1] => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,4,2,1] => 2 = 3 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,1,2,4] => 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => 2 = 3 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
Description
The number of patterns 321 or 3412 in a permutation. A permutation is '''boolean''' if its principal order ideal in the (strong) Bruhat order is boolean. It is shown in [1, Theorem 5.3] that a permutation is boolean if and only if it avoids the two patterns 321 and 3412.
Matching statistic: St000001
Mp00080: Set partitions to permutationPermutations
Mp00310: Permutations toric promotionPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St000001: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [1,3,2] => [3,1,2] => 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => [1,3,2] => 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [1,2,3] => 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [3,2,1] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,1,3] => [2,1,4,3] => 2
{{1,2,3},{4}}
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,4,3,2] => [4,3,1,2] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [4,1,3,2] => [4,1,3,2] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,1,2,3] => [1,2,4,3] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,4,3] => [2,4,1,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [2,3,1,4] => [2,3,1,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,2,4,3] => [4,1,2,3] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,3,2,4] => [3,1,2,4] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,3,4,1] => [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,1,3] => [4,2,1,3] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [3,1,4,2] => [1,3,4,2] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [3,1,2,4] => [1,3,2,4] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,3,1,2] => [1,4,3,2] => 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,2,3,1] => [2,4,3,1] => 3
Description
The number of reduced words for a permutation. This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
The following 116 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000078The number of alternating sign matrices whose left key is the permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St001396Number of triples of incomparable elements in a finite poset. St001684The reduced word complexity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000961The shifted major index of a permutation. St000462The major index minus the number of excedences of a permutation. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001118The acyclic chromatic index of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001933The largest multiplicity of a part in an integer partition. St000668The least common multiple of the parts of the partition. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001128The exponens consonantiae of a partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001857The number of edges in the reduced word graph of a signed permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001943The sum of the squares of the hook lengths of an integer partition. St001875The number of simple modules with projective dimension at most 1. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000181The number of connected components of the Hasse diagram for the poset. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001890The maximum magnitude of the Möbius function of a poset. St000102The charge of a semistandard tableau.