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Your data matches 102 different statistics following compositions of up to 3 maps.
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Matching statistic: St001128
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Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [4]
=> [2,2]
=> [2]
=> 1
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,4,5] => [5]
=> [2,2,1]
=> [2,1]
=> 2
[1,2,3,5,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,4,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,4,5,3] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,5,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,1,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [4,1]
=> [3,2]
=> [2]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St001570
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Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 - 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 2 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 1 - 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,2,3,4] => ([],4)
=> ([],1)
=> ? = 1 + 2
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 2 + 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 1 + 2
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001603
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> ([],1)
=> [1]
=> ? = 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1]
=> ? = 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> [1]
=> ? = 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> ([],1)
=> [1]
=> ? = 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1]
=> ? = 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> [1]
=> ? = 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> ([],1)
=> [1]
=> ? = 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1]
=> ? = 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> [1]
=> ? = 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? = 2
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001629
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00154: Graphs —core⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,2,3,4] => ([],4)
=> ([],1)
=> [1] => ? = 1 - 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1] => ? = 2 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 0 = 1 - 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 0 = 1 - 1
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> [1] => ? = 1 - 1
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 1 - 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => ? = 2 - 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001195
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,3,1,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,2,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,1,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,2,4,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,4,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,2,4,5,3,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,4,5,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,4,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,3,2,6,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,6,5,2,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,6,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
[1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 2
[1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,3,1,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,2,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,4,1,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1
[1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
[1,2,4,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,4,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
[1,2,4,5,3,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,4,5,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,4,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
[1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
[1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
[1,3,2,6,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,3,6,5,2,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,4,2,6,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
Description
The 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 $K[x]/(x^n)$.
Matching statistic: St001001
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,3,1,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[5,1,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,1,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,1,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,1,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,2,1,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,2,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,2,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,3,1,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,4,1,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,4,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
[1,2,4,5,3,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
[1,3,2,6,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,6,2,5,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,6,5,2,4] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,4,2,6,5,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001568The smallest positive integer that does not appear twice in the partition. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001520The number of strict 3-descents. St000068The number of minimal elements in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001256Number of simple reflexive modules that are 2-stable reflexive. St001890The maximum magnitude of the Möbius function of a poset. St001549The number of restricted non-inversions between exceedances. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001895The oddness of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St001964The interval resolution global dimension of a poset. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001429The number of negative entries in a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001863The number of weak excedances of a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice.
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