Identifier
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 2
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [2,3,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 3
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 3
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 1
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,3,2] => [3,4,2,1] => [3,4,2,1] => 2
[4,2,1,3] => [2,4,3,1] => [2,4,3,1] => 2
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 3
[4,3,1,2] => [4,2,3,1] => [4,2,3,1] => 1
[4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 2
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 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
parking function
Description
Interpret the permutation as a parking function.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.