- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000185: Integer partitions ⟶ ℤ

Values

[1,0] => [] => [] => 0

[1,0,1,0] => [1] => [1] => 0

[1,1,0,0] => [] => [] => 0

[1,0,1,0,1,0] => [2,1] => [3] => 0

[1,0,1,1,0,0] => [1,1] => [2] => 0

[1,1,0,0,1,0] => [2] => [1,1] => 1

[1,1,0,1,0,0] => [1] => [1] => 0

[1,1,1,0,0,0] => [] => [] => 0

[1,0,1,0,1,0,1,0] => [3,2,1] => [5,1] => 1

[1,0,1,0,1,1,0,0] => [2,2,1] => [2,2,1] => 4

[1,0,1,1,0,0,1,0] => [3,1,1] => [4,1] => 1

[1,0,1,1,0,1,0,0] => [2,1,1] => [2,2] => 2

[1,0,1,1,1,0,0,0] => [1,1,1] => [2,1] => 1

[1,1,0,0,1,0,1,0] => [3,2] => [5] => 0

[1,1,0,0,1,1,0,0] => [2,2] => [4] => 0

[1,1,0,1,0,0,1,0] => [3,1] => [2,1,1] => 3

[1,1,0,1,0,1,0,0] => [2,1] => [3] => 0

[1,1,0,1,1,0,0,0] => [1,1] => [2] => 0

[1,1,1,0,0,0,1,0] => [3] => [1,1,1] => 3

[1,1,1,0,0,1,0,0] => [2] => [1,1] => 1

[1,1,1,0,1,0,0,0] => [1] => [1] => 0

[1,1,1,1,0,0,0,0] => [] => [] => 0

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => [7,3] => 3

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [3,2,2,2] => 12

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => [6,3] => 3

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [3,2,2,1] => 9

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [4,1,1,1] => 6

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => [7,2] => 2

[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [6,2] => 2

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [4,4] => 4

[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [5,2] => 2

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [4,1,1] => 3

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [3,2,1,1] => 7

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [3,3] => 3

[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [3,1,1] => 3

[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [3,1] => 1

[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => [7,1,1] => 3

[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [2,2,2,2] => 12

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [6,1,1] => 3

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [2,2,2,1] => 9

[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [2,2,2] => 6

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [7,1] => 1

[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [6,1] => 1

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [5,1,1] => 3

[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [5,1] => 1

[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [2,2,1] => 4

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [3,1,1,1] => 6

[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [4,1] => 1

[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [2,2] => 2

[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [2,1] => 1

[1,1,1,0,0,0,1,0,1,0] => [4,3] => [7] => 0

[1,1,1,0,0,0,1,1,0,0] => [3,3] => [6] => 0

[1,1,1,0,0,1,0,0,1,0] => [4,2] => [2,2,1,1] => 7

[1,1,1,0,0,1,0,1,0,0] => [3,2] => [5] => 0

[1,1,1,0,0,1,1,0,0,0] => [2,2] => [4] => 0

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [2,1,1,1] => 6

[1,1,1,0,1,0,0,1,0,0] => [3,1] => [2,1,1] => 3

[1,1,1,0,1,0,1,0,0,0] => [2,1] => [3] => 0

[1,1,1,0,1,1,0,0,0,0] => [1,1] => [2] => 0

[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1] => 6

[1,1,1,1,0,0,0,1,0,0] => [3] => [1,1,1] => 3

[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1] => 1

[1,1,1,1,0,1,0,0,0,0] => [1] => [1] => 0

[1,1,1,1,1,0,0,0,0,0] => [] => [] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [9,5,1] => 7

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => [8,5,1] => 7

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => [3,3,3,1,1] => 16

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [5,2,1,1,1] => 11

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [5,4] => 4

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => [4,2,2,2,1] => 16

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => [7,4,1] => 6

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => [6,2,2] => 6

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => [3,3,2,2] => 13

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => [3,3,3] => 9

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [5,3] => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => [5,5,1] => 7

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => [7,2,1] => 4

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => [5,2,1,1] => 7

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => [4,2,2,1,1] => 13

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => [6,2,1] => 4

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [3,3,1,1] => 8

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [4,3] => 3

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [4,2,1,1,1] => 11

[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [5,2,1] => 4

[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [3,2,2] => 6

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [4,2] => 2

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [3,2] => 2

[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => [9,5] => 5

[1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => [8,5] => 5

[1,1,0,0,1,0,1,1,0,1,0,0] => [4,3,3,2] => [3,3,2,2,2] => 21

[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => [6,6] => 6

[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => [3,3,2,2,1] => 17

[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => [6,1,1,1,1] => 10

[1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => [7,4] => 4

[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => [5,5] => 5

[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => [5,1,1,1,1] => 10

[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [5,1,1,1] => 6

[1,1,0,1,0,0,1,1,0,1,0,0] => [4,3,3,1] => [3,2,2,2,2] => 20

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => [3,2,2,2,1] => 16

[1,1,0,1,0,1,0,0,1,1,0,0] => [4,4,2,1] => [8,3] => 3

[1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => [7,3] => 3

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Description

The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].

Map

Loehr-Warrington

Description

Return a partition whose diagonal inversion number is the length of the preimage.

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 $(n-1,\ldots,2,1)$.