Identifier
Values
[(1,2)] => [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [4,2,1,3] => [2,4,3,1] => 1
[(1,3),(2,4)] => [3,4,1,2] => [1,3,4,2] => [1,4,2,3] => 1
[(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [6,4,2,1,3,5] => [6,5,3,2,4,1] => 2
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [6,1,3,4,2,5] => [3,4,6,5,2,1] => 2
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [6,4,3,2,1,5] => [3,4,2,6,5,1] => 2
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => [6,4,1,3,2,5] => 2
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [6,3,5,2,4,1] => [5,4,2,3,6,1] => 2
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [2,4,6,5,3,1] => [6,1,2,4,5,3] => 1
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [2,5,4,6,1,3] => [5,1,2,6,3,4] => 2
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,6,2,4] => [5,2,1,3,6,4] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => 2
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [6,5,4,2,1,3] => [6,3,4,2,5,1] => 2
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [6,3,5,1,4,2] => [6,2,3,5,4,1] => 2
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [6,1,4,5,3,2] => [5,3,4,6,2,1] => 2
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [1,5,6,4,3,2] => [1,4,6,2,5,3] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 2
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [8,6,1,3,4,2,5,7] => [6,2,8,7,5,4,3,1] => 3
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => [1,2,3,8,4,5,6,7] => 1
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => 2
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [2,7,6,8,5,4,1,3] => [5,7,1,2,8,3,6,4] => 2
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => 3
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
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A cycle peak of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a cycle valley is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
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.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).