- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001564: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [3,2,2,1] => [2,2,1] => 30

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [4,1,1,1] => [1,1,1] => 10

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

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

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

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

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

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [3,2,1,1] => [2,1,1] => 30

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

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

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

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

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

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

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [2,2,2,1] => [2,2,1] => 30

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

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

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

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

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

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

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [3,1,1,1] => [1,1,1] => 10

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

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

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

[1,1,1,0,0,1,0,0,1,0] => [4,2] => [2,2,1,1] => [2,1,1] => 30

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [2,1,1,1] => [1,1,1] => 10

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

[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1] => [1,1,1] => 10

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [9,5,1] => [5,1] => 6

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => [8,5,1] => [5,1] => 6

[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => [7,2,2,2] => [2,2,2] => 10

[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => [6,2,1,1,1,1] => [2,1,1,1,1] => 630

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => [9,2,2,1] => [2,2,1] => 30

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => [6,2,2,1,1,1] => [2,2,1,1,1] => 1260

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

[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => [6,2,2,1,1] => [2,2,1,1] => 210

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => [6,2,1,1,1] => [2,1,1,1] => 140

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => [7,2,2,1] => [2,2,1] => 30

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [5,2,1,1,1] => [2,1,1,1] => 140

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => [9,4,1] => [4,1] => 6

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => [8,4,1] => [4,1] => 6

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => [4,2,2,2,1] => [2,2,2,1] => 140

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

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

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

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

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

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => [6,2,2,1] => [2,2,1] => 30

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => [3,3,2,2] => [3,2,2] => 30

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

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

[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => [9,2,1] => [2,1] => 6

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => [8,2,1] => [2,1] => 6

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

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

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => [5,2,1,1] => [2,1,1] => 30

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => [4,2,2,1,1] => [2,2,1,1] => 210

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

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [3,3,1,1] => [3,1,1] => 30

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

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [4,2,1,1,1] => [2,1,1,1] => 140

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

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

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

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

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

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

[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => [6,1,1,1,1,1] => [1,1,1,1,1] => 126

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

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

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

[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => [6,1,1,1,1] => [1,1,1,1] => 35

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

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

[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => [5,1,1,1,1] => [1,1,1,1] => 35

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

[1,1,0,1,0,0,1,0,1,0,1,0] => [5,4,3,1] => [9,2,1,1] => [2,1,1] => 30

[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,3,1] => [8,2,1,1] => [2,1,1] => 30

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => [3,2,2,2,1] => [2,2,2,1] => 140

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

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

[1,1,0,1,0,1,0,1,0,0,1,0] => [5,3,2,1] => [7,2,1,1] => [2,1,1] => 30

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

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

[1,1,0,1,0,1,1,0,0,0,1,0] => [5,2,2,1] => [3,3,2,1,1] => [3,2,1,1] => 420

>>> Load all 1091 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The value of the forgotten symmetric functions when all variables set to 1.
Let $f_\lambda(x)$ denote the forgotten symmetric functions.
Then the statistic associated with $\lambda$, where $\lambda$ has $\ell$ parts,
is $f_\lambda(1,1,\dotsc,1)$ where there are $\ell$ variables substituted by $1$.

Map

Loehr-Warrington

Description

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

Map

first row removal

Description

Removes the first entry of an integer 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 $(n-1,\ldots,2,1)$.