Identifier
Values
[(1,3),(2,4),(5,6),(7,8),(9,10)] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 2
[(1,4),(2,3),(5,6),(7,8),(9,10)] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 2
[(1,5),(2,3),(4,6),(7,8),(9,10)] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [2,2] => 2
[(1,6),(2,3),(4,5),(7,8),(9,10)] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [2,2] => 2
[(1,7),(2,3),(4,5),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,8),(2,3),(4,5),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,3),(2,5),(4,6),(7,8),(9,10)] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [2,2] => 2
[(1,3),(2,6),(4,5),(7,8),(9,10)] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [2,2] => 2
[(1,3),(2,7),(4,5),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,3),(2,8),(4,5),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,3),(2,8),(4,6),(5,7),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,3),(2,7),(4,6),(5,8),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,3),(2,6),(4,7),(5,8),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,3),(2,5),(4,7),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,8),(2,3),(4,6),(5,7),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,7),(2,3),(4,6),(5,8),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,6),(2,3),(4,7),(5,8),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,5),(2,3),(4,7),(6,8),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,4),(2,3),(5,7),(6,8),(9,10)] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => 1
[(1,3),(2,4),(5,7),(6,8),(9,10)] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => 1
[(1,3),(2,4),(5,8),(6,7),(9,10)] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => 1
[(1,4),(2,3),(5,8),(6,7),(9,10)] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [2,1] => 1
[(1,5),(2,3),(4,8),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,6),(2,3),(4,8),(5,7),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,7),(2,3),(4,8),(5,6),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,8),(2,3),(4,7),(5,6),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,3),(2,5),(4,8),(6,7),(9,10)] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [3] => 1
[(1,3),(2,6),(4,8),(5,7),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,3),(2,7),(4,8),(5,6),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,3),(2,8),(4,7),(5,6),(9,10)] => [1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [2,1] => 1
[(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)] => [1,1,0,1,1,1,0,0,0,0,1,0] => [[4,4,4],[3,1]] => [3,1] => 1
[(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => 1
[(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 6
[(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => [2,1] => 1
[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)] => [1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [3] => 1
[(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)] => [1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => [4] => 1
[(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)] => [1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [3,1] => 1
[(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)] => [1,1,1,0,0,1,0,1,0,0,1,0] => [[4,4,2],[3]] => [3] => 1
[(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)] => [1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [2,2] => 2
[(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => 1
[(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)] => [1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [3,3] => 4
[(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)] => [1,1,1,1,0,1,0,0,0,0,1,0] => [[4,4,4],[3]] => [3] => 1
[(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)] => [1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4],[3]] => [3] => 1
[(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)] => [1,1,1,0,0,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2]] => [2,2] => 2
[(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)] => [1,1,0,0,1,1,0,1,1,0,0,0] => [[4,4,2],[2,1]] => [2,1] => 1
[(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)] => [1,1,0,1,1,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1,1]] => [2,1,1] => 3
[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 2
[(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)] => [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] => 6
[(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 2
[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] => [1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]] => [2,1] => 1
[(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [3,2] => 2
[(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1,1]] => [2,1,1] => 3
[(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)] => [1,1,1,0,0,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1]] => [2,1] => 1
[(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 2
[(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => 2
[(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)] => [1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 16
[(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)] => [1,1,1,1,0,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1]] => [2,1] => 1
[(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)] => [1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3],[2,2]] => [2,2] => 2
[(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1]] => [1,1,1] => 2
[(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)] => [1,1,0,0,1,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1,1]] => [1,1,1] => 2
[(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)] => [1,1,0,1,1,0,0,1,0,0,1,0] => [[4,4,3],[3,1]] => [3,1] => 1
[(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 3
[(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => 1
[(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)] => [1,1,0,1,1,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2,1]] => [2,2,1] => 6
[(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)] => [1,1,0,1,1,0,0,0,1,1,0,0] => [[4,3,3],[2,1]] => [2,1] => 1
[(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1,1]] => [1,1,1] => 2
[(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [2,1] => 1
[(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)] => [1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [3,2] => 2
[(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)] => [1,1,1,1,0,0,0,1,0,0,1,0] => [[4,4,3],[3]] => [3] => 1
[(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1]] => [2,1] => 1
[(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2]] => [2,2] => 2
[(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)] => [1,1,0,1,0,0,1,1,1,0,0,0] => [[4,4,3],[2,2]] => [2,2] => 2
[(1,12),(2,3),(4,5),(6,7),(8,10),(9,11)] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => 1
[(1,11),(2,3),(4,5),(6,7),(8,10),(9,12)] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => 1
[(1,10),(2,3),(4,5),(6,7),(8,11),(9,12)] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => 1
[(1,12),(2,3),(4,5),(6,8),(7,10),(9,11)] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => 2
[(1,11),(2,3),(4,5),(6,8),(7,10),(9,12)] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => 2
[(1,10),(2,3),(4,5),(6,8),(7,11),(9,12)] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => 2
[(1,12),(2,3),(4,6),(5,7),(8,10),(9,11)] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => 1
[(1,11),(2,3),(4,6),(5,7),(8,10),(9,12)] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => 1
[(1,10),(2,3),(4,6),(5,7),(8,11),(9,12)] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => 1
[(1,12),(2,3),(4,6),(5,8),(7,10),(9,11)] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 2
[(1,11),(2,3),(4,6),(5,8),(7,10),(9,12)] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 2
[(1,10),(2,3),(4,6),(5,8),(7,11),(9,12)] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 2
[(1,12),(2,3),(4,7),(5,8),(6,10),(9,11)] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => 1
[(1,11),(2,3),(4,7),(5,8),(6,10),(9,12)] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => 1
[(1,10),(2,3),(4,7),(5,8),(6,11),(9,12)] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => 1
[(1,3),(2,12),(4,5),(6,7),(8,10),(9,11)] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [3] => 1
[(1,3),(2,12),(4,5),(6,8),(7,10),(9,11)] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2] => 2
[(1,3),(2,12),(4,6),(5,7),(8,10),(9,11)] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,1] => 1
[(1,3),(2,12),(4,6),(5,8),(7,10),(9,11)] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 2
[(1,3),(2,12),(4,7),(5,8),(6,10),(9,11)] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,1] => 1
search for individual values
searching the database for the individual values of this statistic
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.
Map
to Dyck path
Description
The Dyck path corresponding to the opener-closer sequence of the perfect matching.
Map
inner shape
Description
The inner shape of a skew partition.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.