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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St001433
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001433: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001433: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 4
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 4
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 4
[1,2,3,4] => [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] => [1,2,4,3] => 6
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 6
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 4
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 6
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 8
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 6
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 6
[2,4,3,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 6
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 8
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 8
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 6
[3,4,2,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 6
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2
[4,1,3,2] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 4
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 4
[4,2,3,1] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 4
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 8
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 6
Description
The flag major index of a signed permutation.
The flag major index of a signed permutation $\sigma$ is:
$$\operatorname{fmaj}(\sigma)=\operatorname{neg}(\sigma)+2\cdot \sum_{i\in \operatorname{Des}_B(\sigma)}{i} ,$$
where $\operatorname{Des}_B(\sigma)$ is the $B$-descent set of $\sigma$; see [1, Eq.(10)].
This statistic is equidistributed with the $B$-inversions ([[St001428]]) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]).
Matching statistic: St000830
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
St000830: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 0
[1,2] => 0
[2,1] => 2
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 4
[3,1,2] => 4
[3,2,1] => 4
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 4
[1,4,2,3] => 4
[1,4,3,2] => 4
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 4
[2,3,4,1] => 6
[2,4,1,3] => 6
[2,4,3,1] => 6
[3,1,2,4] => 4
[3,1,4,2] => 6
[3,2,1,4] => 4
[3,2,4,1] => 6
[3,4,1,2] => 8
[3,4,2,1] => 8
[4,1,2,3] => 6
[4,1,3,2] => 6
[4,2,1,3] => 6
[4,2,3,1] => 6
[4,3,1,2] => 8
[4,3,2,1] => 8
Description
The total displacement of a permutation.
This is, for a permutation $\pi$ of $n$, given by $\sum_{i = 1}^n | \pi(i) - i |.$
This is twice the statistic [[St000029]] and can be found in [3, Problem 5.1.1.28] and also in [1, 2].
Matching statistic: St000422
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8,8,8,8}
[2,4,1,3] => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8,8,8,8}
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8,8,8,8}
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 4
[3,2,4,1] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8,8,8,8}
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,1,2,3] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8,8,8,8}
[4,1,3,2] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8,8,8,8}
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,4,6,6,6,6,6,6,8,8,8,8}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000656
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000656: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 40%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000656: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 40%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> ? = 0
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ? = 0
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? ∊ {0,4}
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 2
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {0,4}
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {0,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Matching statistic: St000799
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000799: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000799: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Values
[1] => [[1]]
=> [(1,2)]
=> [2,1] => 0
[1,2] => [[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 4
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
Description
The number of occurrences of the vincular pattern |213 in a permutation.
This is the number of occurrences of the pattern $(2,1,3)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St001332
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 0
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 4
[3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 4
[3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 4
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
Description
The number of steps on the non-negative side of the walk associated with the permutation.
Consider the walk taking an up step for each ascent, and a down step for each descent of the permutation. Then this statistic is the number of steps that begin and end at non-negative height.
Matching statistic: St001685
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Values
[1] => [[1]]
=> [(1,2)]
=> [2,1] => 0
[1,2] => [[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 0
[2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 4
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8}
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000209
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 5 = 4 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
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: St000503
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [(1,2)]
=> {{1,2}}
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> {{1,4},{2,3}}
=> 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> 5 = 4 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 5 = 4 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 5 = 4 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> {{1,4},{2,3},{5,8},{6,7}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> {{1,8},{2,3},{4,7},{5,6}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> {{1,8},{2,3},{4,7},{5,6}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {0,2,2,2,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,8,8,8,8} + 1
Description
The maximal difference between two elements in a common block.
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000730The maximal arc length of a set partition. St000956The maximal displacement of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000264The girth of a graph, which is not a tree. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. 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. 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 $eA$. 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 $eA$. 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 $eA$. St001557The number of inversions of the second entry of a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001811The Castelnuovo-Mumford regularity of a permutation. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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