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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St000936
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-1,-2,-3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,-4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-1,3,2,-4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000455
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 44%●distinct values known / distinct values provided: 17%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 44%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[1,2,-3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[1,-2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[1,-2,-3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[-1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[-1,2,-3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[-1,-2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[-1,-2,-3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,2,3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,2,-3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,2,-3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,-2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,-2,3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,-2,-3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,-2,-3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,2,3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,2,-3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,2,-3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,-2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,-2,3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,-2,-3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[-1,-2,-3,-4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,2,4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,2,-4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,2,-4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-2,4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-2,-4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,2,4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,2,-4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,-2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,-2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,-3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,-3,2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,-3,-2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,3,2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,3,-2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,3,-2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,-3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,-3,2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,-3,-2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[-1,-3,-2,-4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,-4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,-4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-3,4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-3,-4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,-3,-4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,3,4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,3,-4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,3,-4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-3,4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-3,-4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[-1,-3,-4,-2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,-3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,1,4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,1,-4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,1,-4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,-1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,-1,4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,-1,-4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,-1,-4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,1,4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,1,-4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,1,-4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,-1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,-1,4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,-1,-4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-2,-1,-4,-3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,1,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,1,4,-2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,1,-4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,1,-4,-2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,-1,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,-1,4,-2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,-1,-4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[3,-1,-4,-2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-3,1,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[-3,1,4,-2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001771
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-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] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,-2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,-2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,-2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,-2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[-1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[-1,3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-1,2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-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] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,-2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,-2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,-2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,-2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[-1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[-1,3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-1,2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-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] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,-2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,-4,3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[-1,-2,-4,-3] => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,-2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,-2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,-2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,-3,-2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[-1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[-1,3,2,-4] => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,-5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[-1,-2,-3,-5,-4] => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,-4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,-3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,-2,-4,-3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[-1,2,4,3,-5] => [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St000068
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[-1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,4,1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,4,-1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,4,-1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,-4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,-4,1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,-4,-1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[3,-4,-1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,4,1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,4,-1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,4,-1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,-4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,-4,1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,-4,-1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[-3,-4,-1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,-2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,5,-3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001889
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001889: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 17%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001889: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[1,2,-3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[1,-2,3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[1,-2,-3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[-1,2,3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[-1,2,-3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[-1,-2,3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => [2,3,-1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[-1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,-2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[-1,-2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[-1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
[-1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0
Description
The size of the connectivity set of a signed permutation.
According to [1], the connectivity set of a signed permutation $w\in\mathfrak H_n$ is $n$ minus the number of generators appearing in any reduced word for $w$.
The connectivity set can be defined for arbitrary Coxeter systems. For permutations, see [[St000234]]. For the number of connected elements in a Coxeter system see [[St001888]].
Matching statistic: St001625
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,-3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,-2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,-2,-3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,2,-3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,-2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,-2,-3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,-2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,-2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,-2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,-2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,-2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,-2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,-2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-1,-2,-3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,2,4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,2,-4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,2,-4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,-2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,-2,4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,-2,-4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,-2,-4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,2,4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,2,-4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,2,-4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,-2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,-2,4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,-2,-4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,3,-2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,-3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,-3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,-3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,-3,-2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[-1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[-1,3,2,-4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,-4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,3,-4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,-3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,-3,4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,-3,-4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,-3,-4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,3,4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,3,-4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,3,-4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,-3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,-3,4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,-3,-4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-2,-3,-4,-1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,4,1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,4,-1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,4,-1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,-4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,-4,1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,-4,-1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,-4,-1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,4,1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,4,-1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,4,-1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,-4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,-4,1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,-4,-1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-2,-4,-1,-3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,1,4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,1,-4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,1,-4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,-1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,-1,4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,-1,-4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,-1,-4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,1,4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,1,-4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,1,-4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,-1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,-1,4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,-1,-4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[-3,-1,-4,-2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[3,4,1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001301
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[-1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-1,-2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[1,-2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[-1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[-1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
Matching statistic: St000908
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,-4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-1,-2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,-3,-2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[-1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[-1,3,2,-4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,-2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,-3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,-3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-1,-2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[1,-2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[-1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
Description
The length of the shortest maximal antichain in a poset.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001857The number of edges in the reduced word graph of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001768The number of reduced words of a signed permutation. St001624The breadth of a lattice. 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. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001330The hat guessing number of a graph. St001754The number of tolerances of a finite lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000080The rank of the poset. St000189The number of elements in the poset. St000307The number of rowmotion orbits 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. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000180The number of chains of a poset. St000100The number of linear extensions of a poset. St001909The number of interval-closed sets of a poset. St000634The number of endomorphisms of a poset. St001645The pebbling number of a connected graph.
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