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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001200
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1,0,1,0]
=> 2
[1,2] => [1,2] => [2]
=> [1,1,0,0,1,0]
=> 2
[2,1] => [2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3
[2,1,3] => [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,2] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,1,3,4] => [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,3,4,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,4,1,3] => [3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,2,4,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,4,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,1,2,3] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[4,1,3,2] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001603
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
=> []
=> ?
=> ? = 2 - 2
[1,2] => [2]
=> []
=> ?
=> ? = 2 - 2
[2,1] => [1,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,3] => [3]
=> []
=> ?
=> ? = 2 - 2
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 3 - 2
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 3 - 2
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 2 - 2
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2 - 2
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 2 - 2
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2 - 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 3 - 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2 - 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? = 2 - 2
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[3,2,1,7,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,1,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,1,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,5,1,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,7,2,1,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,7,2,6,1,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,7,2,6,5,1,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,1,7,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,7,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,7,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,7,2,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,7,2,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,7,3,2,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,7,3,2,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,7,6,5,8,4] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,1,7,6,8,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,4,1,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,4,3,2,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,4,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,5,1,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,5,4,3,2,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,7,6,5,4,8] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,1,8,5,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,1,8,3,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,8,2,5,1,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,7,2,6,8,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,8,1,4,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,4,3,2,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,4,3,8,2,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,8,3,2,5,1,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,8,5,7,1,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,4,8,3,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,5,4,8,3,2,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,8,4,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,8,4,7,6,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,1,3,8,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,1,5,4,8] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,7,3,2,6,8,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,8,1,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,1,8,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,4,8,7,1,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,5,8,1,4] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,8,4,3,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,4,3,8,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,4,8,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,8,1,3,7,6,2,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,5,4,3,8,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 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: St001605
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
=> []
=> ?
=> ? = 2 - 2
[1,2] => [2]
=> []
=> ?
=> ? = 2 - 2
[2,1] => [1,1]
=> [1]
=> []
=> ? = 2 - 2
[1,2,3] => [3]
=> []
=> ?
=> ? = 2 - 2
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 3 - 2
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 3 - 2
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 2 - 2
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2 - 2
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 2 - 2
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 2 - 2
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2 - 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 3 - 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2 - 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 2
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? = 2 - 2
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 2
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 3 - 2
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 2
[3,2,1,7,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,1,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,1,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,5,1,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,7,2,1,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,7,2,6,1,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,7,2,6,5,1,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,1,7,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,7,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,7,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,7,2,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,7,2,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,7,3,2,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,7,3,2,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,7,6,5,8,4] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,1,7,6,8,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,4,1,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,4,3,2,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,4,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,5,1,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,5,4,3,2,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,7,6,5,4,8] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,1,8,5,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,1,8,3,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,8,2,5,1,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,7,2,6,8,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,8,1,4,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,4,3,2,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,4,3,8,2,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,8,3,2,5,1,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,3,2,8,5,7,1,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,4,8,3,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,5,4,8,3,2,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,1,8,4,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,8,4,7,6,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,1,3,8,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,1,5,4,8] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,7,3,2,6,8,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,8,1,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,1,8,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,4,8,7,1,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[3,2,7,6,5,8,1,4] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[5,1,8,4,3,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,4,3,8,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,4,8,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[4,8,1,3,7,6,2,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,5,4,3,8,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 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: St001604
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
=> []
=> ?
=> ? = 2 - 3
[1,2] => [2]
=> []
=> ?
=> ? = 2 - 3
[2,1] => [1,1]
=> [1]
=> []
=> ? = 2 - 3
[1,2,3] => [3]
=> []
=> ?
=> ? = 2 - 3
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 3 - 3
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 3 - 3
[2,3,1] => [2,1]
=> [1]
=> []
=> ? = 2 - 3
[3,1,2] => [2,1]
=> [1]
=> []
=> ? = 2 - 3
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 3
[1,2,3,4] => [4]
=> []
=> ?
=> ? = 2 - 3
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 2 - 3
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? = 2 - 3
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 3
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? = 3 - 3
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 2 - 3
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 3
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? = 3 - 3
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 3
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 3
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 2 - 3
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? = 2 - 3
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? = 3 - 3
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? = 3 - 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? = 3 - 3
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? = 3 - 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? = 3 - 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 3 - 3
[3,2,1,7,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,7,1,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,7,6,1,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,7,6,5,1,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,7,2,1,6,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,7,2,6,1,5,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,7,2,6,5,1,4] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,1,7,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,7,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,7,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,7,2,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,7,2,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,7,3,2,1,6,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,7,3,2,6,1,5] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,1,7,6,5,8,4] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,1,7,6,8,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,4,1,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,3,2,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,1,4,8,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,5,1,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,4,3,2,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,1,7,6,5,4,8] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,1,8,5,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,1,8,3,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,8,2,5,1,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,7,2,6,8,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,8,1,4,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[5,1,4,3,2,8,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[5,1,4,3,8,2,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,8,3,2,5,1,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,3,2,8,5,7,1,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[5,1,4,8,3,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,4,8,3,2,7,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,1,8,4,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,8,4,7,6,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,1,3,8,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,7,6,1,5,4,8] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,7,3,2,6,8,1,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,8,1,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,1,8,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,4,8,7,1,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[3,2,7,6,5,8,1,4] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[5,1,8,4,3,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,3,8,7,2,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,4,8,3,2,7,6,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[4,8,1,3,7,6,2,5] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
[1,5,4,3,8,7,2,6] => [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0 = 3 - 3
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: St000298
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[2,3,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,4,3,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 3 - 1
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,5,1,4,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,5,3,1,4] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,5,4,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[3,1,4,2,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[3,1,4,5,2] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000640
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000640: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000640: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> ? = 2 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2 = 3 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[2,3,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,4,3,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 3 - 1
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,5,1,4,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,5,3,1,4] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,5,4,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[3,1,4,2,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
Description
The rank of the largest boolean interval in a poset.
Matching statistic: St000307
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[2,3,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,4,3,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 3 - 1
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,5,1,4,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001632
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> ? = 2 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,2,3,4] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[2,3,5,1,4] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[2,4,3,5,1] => [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 3 - 1
[2,4,5,1,3] => [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
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