Your data matches 49 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000936
Mp00163: Signed permutations permutationPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00202: Integer partitions first row removalInteger 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 permutationPermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
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.
Mp00163: Signed permutations permutationPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00170: Permutations to signed permutationSigned 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 $0 < \pi(i) < -\pi(j)$.
Mp00163: Signed permutations permutationPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00170: Permutations to signed permutationSigned 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 $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Mp00163: Signed permutations permutationPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00170: Permutations to signed permutationSigned 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.
Mp00163: Signed permutations permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00209: Permutations pattern posetPosets
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 permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
Mp00168: Signed permutations Kreweras complementSigned 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 permutationPermutations
Mp00065: Permutations permutation posetPosets
Mp00195: Posets order idealsLattices
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]].
Mp00163: Signed permutations permutationPermutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00209: Permutations pattern posetPosets
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.
Mp00163: Signed permutations permutationPermutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00209: Permutations pattern posetPosets
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.