Identifier
Values
[(1,3),(2,4)] => [3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 2
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [2,4,6,5,3,1] => [1,1,0,1,1,0,1,1,0,0,0,0] => 1
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0] => 2
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,6,2,4] => [1,1,1,0,0,1,1,0,1,0,0,0] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
search for individual values
searching the database for the individual values of this statistic
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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
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.).