Identifier
-
Mp00038:
Integer compositions
—reverse⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000145: Integer partitions ⟶ ℤ
Values
[1,1] => [1,1] => [1,0,1,0] => [1] => 0
[1,1,1] => [1,1,1] => [1,0,1,0,1,0] => [2,1] => 0
[1,2] => [2,1] => [1,1,0,0,1,0] => [2] => 1
[2,1] => [1,2] => [1,0,1,1,0,0] => [1,1] => -1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 0
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 0
[1,3] => [3,1] => [1,1,1,0,0,0,1,0] => [3] => 2
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => -1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 0
[3,1] => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => -2
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 0
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 0
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 2
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 0
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 0
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 3
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => -1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 0
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => -1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => -2
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => -1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => -3
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 3
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 0
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 4
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 2
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 0
[4,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => -3
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => -2
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => -4
[1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 5
[2,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 3
[5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => -3
[6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => -5
[1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 6
[7,1] => [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] => -6
[1,8] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8] => 7
[8,1] => [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] => -7
[1,9] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [9] => 8
[9,1] => [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] => -8
[1,10] => [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] => 9
[10,1] => [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] => -9
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Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
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 $(n-1,\ldots,2,1)$.
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.
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