Your data matches 190 different statistics following compositions of up to 3 maps.
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Mp00068: Permutations Simion-Schmidt mapPermutations
St000651: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 2
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 3
[1,2,4,3] => [1,4,3,2] => 3
[1,3,2,4] => [1,4,3,2] => 3
[1,3,4,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,4,3] => 3
[2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => 2
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,4,2] => 3
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => 1
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 0
Description
The maximal size of a rise in a permutation. This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Mp00160: Permutations graph of inversionsGraphs
St001120: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The length of a longest path in a graph.
Mp00160: Permutations graph of inversionsGraphs
St001110: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1 = 0 + 1
[1,2] => ([],2)
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The 3-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-dynamic chromatic number of a graph.
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St000141: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 2
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 3
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => 3
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [2,1,4,3] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => 2
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 3
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => 3
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => 1
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 3
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Mp00160: Permutations graph of inversionsGraphs
Mp00117: Graphs Ore closureGraphs
St000171: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Mp00240: Permutations weak exceedance partitionSet partitions
Mp00080: Set partitions to permutationPermutations
St000209: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 2
[3,1,2] => {{1,3},{2}}
=> [3,2,1] => 2
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 1
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,1,3] => {{1,2,4},{3}}
=> [2,4,3,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,2,4,1] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 3
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[4,1,3,2] => {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[4,2,1,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[4,3,1,2] => {{1,4},{2,3}}
=> [4,3,2,1] => 3
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 3
Description
Maximum difference of elements in cycles. Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$. The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000272
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St000272: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000536
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St000536: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The pathwidth of a graph.
Mp00277: Permutations catalanizationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
St001090: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [3,1,2] => 2
[3,1,2] => [2,3,1] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[2,4,1,3] => [4,3,1,2] => [4,2,3,1] => 3
[2,4,3,1] => [2,4,3,1] => [3,4,1,2] => 3
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 2
[3,1,4,2] => [2,3,4,1] => [4,1,2,3] => 3
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [3,2,4,1] => [2,4,1,3] => 2
[3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 3
[3,4,2,1] => [3,4,2,1] => [4,1,3,2] => 3
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 3
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 3
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 2
[4,2,3,1] => [3,4,2,1] => [4,1,3,2] => 3
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 3
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 3
Description
The number of pop-stack-sorts needed to sort a permutation. The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order. A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St001270
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St001270: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The bandwidth of a graph. The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$. We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
The following 180 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000019The cardinality of the support of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001117The game chromatic index of a graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001375The pancake length of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000013The height of a Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000240The number of indices that are not small excedances. St000653The last descent of a permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001497The position of the largest weak excedence of a permutation. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000503The maximal difference between two elements in a common block. St000956The maximal displacement of a permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000539The number of odd inversions of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001480The number of simple summands of the module J^2/J^3. St001721The degree of a binary word. St000485The length of the longest cycle of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000741The Colin de Verdière graph invariant. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001645The pebbling number of a connected graph. St000681The Grundy value of Chomp on Ferrers diagrams. St001596The number of two-by-two squares inside a skew partition. St000444The length of the maximal rise of a Dyck path. 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000259The diameter of a connected graph. St001118The acyclic chromatic index of a graph. St001060The distinguishing index of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St000260The radius of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001733The number of weak left to right maxima of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000455The second largest eigenvalue of a graph if it is integral. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001568The smallest positive integer that does not appear twice in the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000675The number of centered multitunnels of a Dyck path. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000662The staircase size of the code of a permutation. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000264The girth of a graph, which is not a tree. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000091The descent variation of a composition. St000173The segment statistic of a semistandard tableau. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001778The largest greatest common divisor of an element and its image in a permutation. St000422The energy of a graph, if it is integral. St001926Sparre Andersen's position of the maximum of a signed permutation. St000456The monochromatic index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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 $eA$. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled 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. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra.