Identifier
Values
[(1,2)] => [2,1] => [2,1] => [1,1,0,0] => 1
[(1,2),(3,4)] => [2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0] => 1
[(1,3),(2,4)] => [3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 1
[(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(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] => 1
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(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] => 0
[(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] => 1
[(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] => 0
[(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] => 0
[(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] => 1
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [6,3,5,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[(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] => 1
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
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
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
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.).