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Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000938
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
Description
The number of zeros 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 conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St001214
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St001771
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,4,+,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,-,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,-,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,+,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,-,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,+,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,4,1,2,+] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,4,1,2,-] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[3,5,1,+,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,1,-,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[3,5,2,+,1] => [3,5,2,4,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
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 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,4,+,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,-,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,-,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,+,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,-,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,+,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,4,1,2,+] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,4,1,2,-] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[3,5,1,+,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,1,-,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[3,5,2,+,1] => [3,5,2,4,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
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 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,4,+,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,-,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,-,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,+,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,-,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,+,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,4,1,2,+] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,4,1,2,-] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[3,5,1,+,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,1,-,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[3,5,2,+,1] => [3,5,2,4,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => ([(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
[2,4,+,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[2,4,-,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[3,1,4,2] => [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,+,1,+] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,-,1,+] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,+,1,-] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,-,1,-] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => ([(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] => [3,2,4,1] => ([(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] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,1,+,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,1,-,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,+,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,-,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,+,+,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,-,+,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,+,-,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,-,-,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,5,+,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => ([(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)
=> 1 = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[+,4,+,2,+] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,+,2,+] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,4,-,2,+] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,4,+,2,-] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,-,2,+] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,+,2,-] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,4,-,2,-] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,-,2,-] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,4,+,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,4,+,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,4,-,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,4,-,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[-,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[+,4,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,4,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,5,2,+,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[-,5,2,+,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[+,5,2,-,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[-,5,2,-,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
[+,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,5,+,+,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,5,+,+,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,5,-,+,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,5,+,-,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,5,-,+,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,5,+,-,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,5,-,-,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[-,5,-,-,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[+,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[-,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[+,5,4,3,2] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[-,5,4,3,2] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,1,+,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[2,1,-,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[2,1,4,3,+] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[2,1,4,3,-] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[3,+,1,+,+] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,-,1,+,+] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,+,1,-,+] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,+,1,+,-] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,-,1,-,+] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,-,1,+,-] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,+,1,-,-] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,-,1,-,-] => [3,2,1,4,5] => ([(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)
=> 1 = 0 + 1
[3,4,5,2,1] => [3,4,5,2,1] => ([(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)
=> 1 = 0 + 1
[4,3,2,1,+] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,3,2,1,-] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,5,+,2,1] => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4,5,-,2,1] => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001772
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0
[2,4,+,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[2,4,-,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,-,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,+,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,+,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
[2,3,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => ? = 0
[2,3,5,+,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,3,5,-,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => ? = 0
[2,4,+,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,-,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
[2,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => [-2,-4,-5,-3,-1] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,+,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,-,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,+,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,+,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 1
[2,5,4,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
Description
The number of occurrences of the signed pattern 12 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: St001863
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0
[2,4,+,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[2,4,-,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,-,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,+,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,+,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
[2,3,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => ? = 0
[2,3,5,+,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,3,5,-,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => ? = 0
[2,4,+,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,-,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
[2,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => [-2,-4,-5,-3,-1] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,+,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,-,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,+,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,+,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 1
[2,5,4,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Matching statistic: St001864
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0
[2,4,+,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[2,4,-,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,-,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,+,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,+,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
[2,3,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => ? = 0
[2,3,5,+,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,3,5,-,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => ? = 0
[2,4,+,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,-,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
[2,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => [-2,-4,-5,-3,-1] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,+,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,-,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,+,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,+,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 1
[2,5,4,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001867
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [-3,-2,-1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0
[2,4,+,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[2,4,-,1] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 0
[3,+,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,-,4,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0
[4,+,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,+,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,+,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,-,-,1] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 0
[2,3,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => ? = 0
[2,3,5,+,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,3,5,-,1] => [2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => ? = 0
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 0
[2,4,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => ? = 0
[2,4,+,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,+] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,-,1,-] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 0
[2,4,+,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,-,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 0
[2,4,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 0
[2,4,5,3,1] => [2,4,5,3,1] => [2,4,5,3,1] => [-2,-4,-5,-3,-1] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => ? = 1
[2,5,+,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,-,1,4] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 0
[2,5,+,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,+,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,+,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,-,-,1] => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 1
[2,5,4,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => ? = 0
Description
The number of alignments of type EN of a signed permutation.
An alignment of type EN of a signed permutation π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold:
* $-i < 0 < -\pi(i) < \pi(j) < j$
* $i \leq\pi(i) < \pi(j) < j$.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001866The nesting alignments of a signed permutation. St001893The flag descent of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001896The number of right descents of a signed permutations. St001892The flag excedance statistic of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. 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. St001946The number of descents in a parking function. 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. St001851The number of Hecke atoms 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 $K[x]/(x^n)$. St001768The number of reduced words of a signed permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000907The number of maximal antichains of minimal length in a poset. St001490The number of connected components of a skew partition.
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