Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [2,1,4,3] => 0
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 0
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,3,1,5,4] => 0
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,4,1,5,2] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,3,4,1,6,5] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,2,4] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,4,1,5,6,2] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,3,4,6,2,5] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,1,3,5,4] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,2,4] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,2,6,5,3] => 0
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [1,3,4,5,7,2,6] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,1,6,4,3,5] => 0
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,2,5] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,6,1,5,2,4] => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,5,1,6,3] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,4,1,6,2,3] => 1
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [1,4,2,5,7,6,3] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,1,4,6,5] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,6,3,5] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,4,1,2,6,5] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,1,6,3,2,4] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,3,5,4] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,5,2,6,3] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,2,5,7,4,3,6] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,1,6,3,5] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,2,6,4,3,5] => 0
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,1,6,3,4] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [5,3,6,1,2,4] => 1
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,2,3,5,4] => 0
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 1
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => 0
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [1,4,5,7,2,3,6] => 0
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,3,4,6,5] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,2,6,4,5] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,6,3,5] => 0
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,4,2,3,7,6,5] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,6,3,4] => 0
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => [1,3,4,2,7,5,6] => 0
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,4,2,5,7,3,6] => 0
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [1,2,3,5,4,7,6] => 0
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,2,3,6,7,4,5] => 0
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [1,2,4,3,7,5,6] => 0
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,2,3,7,5,4,6] => 0
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,7,3,4,6,5] => 0
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,3,2,4,7,5,6] => 0
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,2,4,7,3,5,6] => 0
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,2,7,3,5,4,6] => 0
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[] => [] => [1] => [1] => 0
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 nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see St000155The number of exceedances (also excedences) of a 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.).
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.