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Matching statistic: St000941
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: 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
St000941: 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]
=> 1
[+,+,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 characters of the symmetric group whose value on the partition is even.
Matching statistic: St001896
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 0 + 2
[3,-,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 0 + 2
[2,4,+,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[2,4,-,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[3,+,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[3,-,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 0 + 2
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 0 + 2
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 0 + 2
[4,+,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[4,-,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[4,+,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[4,-,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2 = 0 + 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 1 + 2
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 0 + 2
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 0 + 2
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 2 = 0 + 2
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 2 = 0 + 2
[+,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[-,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[+,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[-,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 0 + 2
[+,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 0 + 2
[-,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 0 + 2
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2 = 0 + 2
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2 = 0 + 2
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2 = 0 + 2
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0 + 2
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0 + 2
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0 + 2
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0 + 2
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0 + 2
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0 + 2
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0 + 2
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1 + 2
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1 + 2
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1 + 2
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0 + 2
[3,1,4,2,+] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[3,1,4,2,-] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[3,1,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0 + 2
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0 + 2
[3,1,5,+,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[3,1,5,-,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,-,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0 + 2
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1 + 2
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1 + 2
[3,+,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[3,-,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[3,+,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[3,-,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0 + 2
[3,+,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0 + 2
[3,-,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0 + 2
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1 + 2
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1 + 2
[3,+,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[3,-,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[3,+,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[3,-,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 2
[3,4,1,2,+] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0 + 2
[3,4,1,2,-] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0 + 2
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0 + 2
[3,5,1,+,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1 + 2
[3,5,1,-,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1 + 2
[3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1 + 2
[3,5,2,+,1] => [3,5,2,4,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1 + 2
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St001845
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[3,-,1] => [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[+,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
[-,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
[+,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
[-,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 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] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[2,4,-,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 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,+] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
[3,-,1,+] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
[3,+,1,-] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
[3,-,1,-] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
[3,+,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[3,-,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 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
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,1,+,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[4,1,-,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[4,+,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[4,-,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[4,+,+,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,-,+,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,+,-,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,-,-,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
[+,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[-,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[+,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,4,2,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,+,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,-,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,+,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,-,2,+] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,+,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,-,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[-,4,-,2,-] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[+,4,+,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[-,4,+,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[+,4,-,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[-,4,-,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[+,4,5,2,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[-,4,5,2,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[+,4,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 0
[-,4,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 0
[+,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[-,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[+,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[-,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[+,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 0
[2,1,+,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 0
[2,1,-,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 0
[2,1,4,3,+] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 0
[2,1,4,3,-] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 0
[2,4,1,3,+] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 0
[2,4,1,3,-] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 0
[3,1,2,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 0
[3,1,4,2,+] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 0
[3,1,4,2,-] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 0
[+,+,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 0
[-,+,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 0
[+,-,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 0
[-,-,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 0
[+,3,2,+,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 0
[-,3,2,+,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 0
[+,3,2,-,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 0
[-,3,2,-,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 0
[+,3,2,5,4,+] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 0
[-,3,2,5,4,+] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 0
[+,3,2,5,4,-] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 0
[-,3,2,5,4,-] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 0
[2,1,+,+,6,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 0
[2,1,-,+,6,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 0
[2,1,+,-,6,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 0
[2,1,-,-,6,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 0
[2,1,+,5,4,+] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> 0
[2,1,-,5,4,+] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001490
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
[3,+,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 0 + 1
[2,4,+,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 0 + 1
[2,4,-,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
[3,+,1,+] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
[3,-,1,+] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
[3,+,1,-] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
[3,-,1,-] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 0 + 1
[4,1,+,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,1,-,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,+,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,-,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,+,+,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,-,+,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,+,-,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,-,-,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 1 + 1
[+,+,5,+,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[+,4,+,2,+] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[-,4,+,2,+] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[+,4,-,2,+] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[+,4,+,2,-] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[-,4,-,2,+] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[-,4,+,2,-] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[+,4,-,2,-] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[-,4,-,2,-] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[+,4,+,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[-,4,+,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[+,4,-,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[-,4,-,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[+,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[-,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[+,4,5,3,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[-,4,5,3,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[+,5,2,+,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[-,5,2,+,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[+,5,2,-,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[-,5,2,-,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[+,5,+,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[-,5,+,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[2,1,+,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
[2,1,-,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
[2,1,4,3,+] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[2,1,4,3,-] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[2,1,4,5,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 0 + 1
[2,3,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1 = 0 + 1
Description
The number of connected components of a skew partition.
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