Identifier
Values
{{1}} => [1] => [1] => [1] => 1
{{1,2}} => [2,1] => [2,1] => [2,1] => 1
{{1},{2}} => [1,2] => [1,2] => [1,2] => 2
{{1,2,3}} => [2,3,1] => [3,2,1] => [3,2,1] => 1
{{1,2},{3}} => [2,1,3] => [2,1,3] => [2,1,3] => 2
{{1,3},{2}} => [3,2,1] => [2,3,1] => [2,3,1] => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,3,2] => 2
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 3
{{1,2,3,4}} => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 2
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
{{1,3,4},{2}} => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 2
{{1,3},{2,4}} => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 3
{{1,4},{2,3}} => [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3
{{1,4},{2},{3}} => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 3
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
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Description
The number of critical left to right maxima of the parking functions.
An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it.
This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via
$$ \sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P}, $$
where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
parking function
Description
Interpret the permutation as a parking function.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.