- FindStat

Identifier

Mp00305: Permutations —parking function⟶ Parking functions
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
St001898: Ordered set partitions ⟶ ℤ

Values

[1] => [1] => [{1}] => 0

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

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

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

[1,3,2] => [1,3,2] => [{1},{3},{2}] => 1

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

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

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

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

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

[1,2,4,3] => [1,2,4,3] => [{1},{2},{4},{3}] => 2

[1,3,2,4] => [1,3,2,4] => [{1},{3},{2},{4}] => 1

[1,3,4,2] => [1,3,4,2] => [{1},{4},{2},{3}] => 2

[1,4,2,3] => [1,4,2,3] => [{1},{3},{4},{2}] => 2

[1,4,3,2] => [1,4,3,2] => [{1},{4},{3},{2}] => 3

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

[2,1,4,3] => [2,1,4,3] => [{2},{1},{4},{3}] => 2

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

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

[2,4,1,3] => [2,4,1,3] => [{3},{1},{4},{2}] => 1

[2,4,3,1] => [2,4,3,1] => [{4},{1},{3},{2}] => 1

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

[3,1,4,2] => [3,1,4,2] => [{2},{4},{1},{3}] => 1

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

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

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

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

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

[4,1,3,2] => [4,1,3,2] => [{2},{4},{3},{1}] => 1

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

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

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

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

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

[1,2,3,5,4] => [1,2,3,5,4] => [{1},{2},{3},{5},{4}] => 3

[1,2,4,3,5] => [1,2,4,3,5] => [{1},{2},{4},{3},{5}] => 2

[1,2,4,5,3] => [1,2,4,5,3] => [{1},{2},{5},{3},{4}] => 4

[1,2,5,3,4] => [1,2,5,3,4] => [{1},{2},{4},{5},{3}] => 4

[1,2,5,4,3] => [1,2,5,4,3] => [{1},{2},{5},{4},{3}] => 6

[1,3,2,4,5] => [1,3,2,4,5] => [{1},{3},{2},{4},{5}] => 1

[1,3,2,5,4] => [1,3,2,5,4] => [{1},{3},{2},{5},{4}] => 4

[1,3,4,2,5] => [1,3,4,2,5] => [{1},{4},{2},{3},{5}] => 2

[1,3,4,5,2] => [1,3,4,5,2] => [{1},{5},{2},{3},{4}] => 3

[1,3,5,2,4] => [1,3,5,2,4] => [{1},{4},{2},{5},{3}] => 4

[1,3,5,4,2] => [1,3,5,4,2] => [{1},{5},{2},{4},{3}] => 5

[1,4,2,3,5] => [1,4,2,3,5] => [{1},{3},{4},{2},{5}] => 2

[1,4,2,5,3] => [1,4,2,5,3] => [{1},{3},{5},{2},{4}] => 4

[1,4,3,2,5] => [1,4,3,2,5] => [{1},{4},{3},{2},{5}] => 3

[1,4,3,5,2] => [1,4,3,5,2] => [{1},{5},{3},{2},{4}] => 4

[1,4,5,2,3] => [1,4,5,2,3] => [{1},{4},{5},{2},{3}] => 4

[1,4,5,3,2] => [1,4,5,3,2] => [{1},{5},{4},{2},{3}] => 5

[1,5,2,3,4] => [1,5,2,3,4] => [{1},{3},{4},{5},{2}] => 3

[1,5,2,4,3] => [1,5,2,4,3] => [{1},{3},{5},{4},{2}] => 5

[1,5,3,2,4] => [1,5,3,2,4] => [{1},{4},{3},{5},{2}] => 4

[1,5,3,4,2] => [1,5,3,4,2] => [{1},{5},{3},{4},{2}] => 5

[1,5,4,2,3] => [1,5,4,2,3] => [{1},{4},{5},{3},{2}] => 5

[1,5,4,3,2] => [1,5,4,3,2] => [{1},{5},{4},{3},{2}] => 6

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

[2,1,3,5,4] => [2,1,3,5,4] => [{2},{1},{3},{5},{4}] => 3

[2,1,4,3,5] => [2,1,4,3,5] => [{2},{1},{4},{3},{5}] => 2

[2,1,4,5,3] => [2,1,4,5,3] => [{2},{1},{5},{3},{4}] => 4

[2,1,5,3,4] => [2,1,5,3,4] => [{2},{1},{4},{5},{3}] => 4

[2,1,5,4,3] => [2,1,5,4,3] => [{2},{1},{5},{4},{3}] => 6

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

[2,3,1,5,4] => [2,3,1,5,4] => [{3},{1},{2},{5},{4}] => 3

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

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

[2,3,5,1,4] => [2,3,5,1,4] => [{4},{1},{2},{5},{3}] => 2

[2,3,5,4,1] => [2,3,5,4,1] => [{5},{1},{2},{4},{3}] => 2

[2,4,1,3,5] => [2,4,1,3,5] => [{3},{1},{4},{2},{5}] => 1

[2,4,1,5,3] => [2,4,1,5,3] => [{3},{1},{5},{2},{4}] => 3

[2,4,3,1,5] => [2,4,3,1,5] => [{4},{1},{3},{2},{5}] => 1

[2,4,3,5,1] => [2,4,3,5,1] => [{5},{1},{3},{2},{4}] => 1

[2,4,5,1,3] => [2,4,5,1,3] => [{4},{1},{5},{2},{3}] => 2

[2,4,5,3,1] => [2,4,5,3,1] => [{5},{1},{4},{2},{3}] => 2

[2,5,1,3,4] => [2,5,1,3,4] => [{3},{1},{4},{5},{2}] => 2

[2,5,1,4,3] => [2,5,1,4,3] => [{3},{1},{5},{4},{2}] => 4

[2,5,3,1,4] => [2,5,3,1,4] => [{4},{1},{3},{5},{2}] => 2

[2,5,3,4,1] => [2,5,3,4,1] => [{5},{1},{3},{4},{2}] => 2

[2,5,4,1,3] => [2,5,4,1,3] => [{4},{1},{5},{3},{2}] => 3

[2,5,4,3,1] => [2,5,4,3,1] => [{5},{1},{4},{3},{2}] => 3

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

[3,1,2,5,4] => [3,1,2,5,4] => [{2},{3},{1},{5},{4}] => 3

[3,1,4,2,5] => [3,1,4,2,5] => [{2},{4},{1},{3},{5}] => 1

[3,1,4,5,2] => [3,1,4,5,2] => [{2},{5},{1},{3},{4}] => 2

[3,1,5,2,4] => [3,1,5,2,4] => [{2},{4},{1},{5},{3}] => 3

[3,1,5,4,2] => [3,1,5,4,2] => [{2},{5},{1},{4},{3}] => 4

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

[3,2,1,5,4] => [3,2,1,5,4] => [{3},{2},{1},{5},{4}] => 3

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

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

[3,2,5,1,4] => [3,2,5,1,4] => [{4},{2},{1},{5},{3}] => 2

[3,2,5,4,1] => [3,2,5,4,1] => [{5},{2},{1},{4},{3}] => 2

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

[3,4,1,5,2] => [3,4,1,5,2] => [{3},{5},{1},{2},{4}] => 1

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

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

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

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

[3,5,1,2,4] => [3,5,1,2,4] => [{3},{4},{1},{5},{2}] => 1

[3,5,1,4,2] => [3,5,1,4,2] => [{3},{5},{1},{4},{2}] => 2

>>> Load all 720 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of occurrences of an 132 pattern in an ordered set partition.
An occurrence of a pattern $\pi\in\mathfrak S_k$ ordered set partition with blocks $B_1|\dots|B_\ell$ is a sequence of elements $e_1,\dots,e_k$ with $e_i\in B_{j_i}$ and $j_1 < \dots < j_k$ order-isomorphic to $\pi$.

Map

parking function

Description

Interpret the permutation as a parking function.

Map

to ordered set partition

Description

Return the ordered set partition whose blocks are the labels of the runs of the labelled Dyck path.