Identifier
Values
[1] => [1,0] => [] => 1
[1,1] => [1,0,1,0] => [1] => 2
[2] => [1,1,0,0] => [] => 1
[1,1,1] => [1,0,1,0,1,0] => [2,1] => 5
[1,2] => [1,0,1,1,0,0] => [1,1] => 3
[2,1] => [1,1,0,0,1,0] => [2] => 3
[3] => [1,1,1,0,0,0] => [] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 14
[1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 9
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 10
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 4
[2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 9
[2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 6
[3,1] => [1,1,1,0,0,0,1,0] => [3] => 4
[4] => [1,1,1,1,0,0,0,0] => [] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 42
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 28
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 32
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 14
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 32
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 22
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 17
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 28
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 19
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 22
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 10
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 14
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 10
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 5
[5] => [1,1,1,1,1,0,0,0,0,0] => [] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 20
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 26
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 6
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 15
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 20
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 20
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 15
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 6
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [] => 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 7
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => 21
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 21
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 7
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 8
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 8
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => 9
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8] => 9
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [] => 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => 10
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [9] => 10
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [] => 1
[1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1] => 11
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10] => 11
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Description
The number of partitions contained in the given partition.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
bounce path
Description
The bounce path determined by an integer composition.