- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St000318The number of addable cells of the Ferrers diagram of an integer partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 16514 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than St000159The number of distinct parts of the integer partition., the number of distinct parts of the partition.

Map

to car permutation

Description

Maps a parking function to the permutation of cars that correspond to that parking function.
For example, the image $[2, 4, 5, 6, 3, 1, 7]$ means that car 2 takes spots 1, car 4 takes spot 2, and so on.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

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