Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001605
Mp00233: Dyck paths skew partitionSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001605: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 6
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 16
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[5,5,5],[4,3]]
=> [4,3]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[5,5,5],[3,3]]
=> [3,3]
=> [3]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[4,4,4,4],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [[4,4,4,4],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 6
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [[4,4,4,3],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[4,4,4,4],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [[4,4,4,4],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[4,4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,1,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,1],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2,1],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2,1],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2,1],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
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: St001571
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
Mp00030: Dyck paths zeta mapDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St001571: Integer partitions ⟶ ℤResult quality: 3% values known / values provided: 11%distinct values known / distinct values provided: 3%
Values
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1,1]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 6
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1,1]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 16
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 6
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,1]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,3,2,1]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,2,1]
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1,1,1,1]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,2,1,1,1]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,2,1]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,1,1]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,1,1,1]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,2,2,1]
=> ? = 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1,1,1]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? = 24
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2,1]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1]
=> 1
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.