Identifier
Values
[1,0] => [2,1] => [2,1] => 0
[1,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,1,0,0] => [2,3,1] => [2,3,1] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [2,1,4,3] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [1,4,3,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,2,4] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,4,1,5,2] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,2,4] => 0
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,5,1,2] => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,3,1,5,4] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,5,1,4,3] => 0
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,3,5,4,2] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,6,3,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,6,3,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,2,3,5,4] => 0
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,5,2,6,3] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,2,6,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [5,3,6,1,2,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,2,6,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,4,1,6,2,3] => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,4,1,2,6,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,6,1,5,2,4] => 0
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,2,6,5,3] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,4,1,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,3,4,6,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [5,2,1,6,3,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,1,6,3,5] => 1
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,3,5,4] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,5,1,6,3] => 0
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,1,6,3,2,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,6,3,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,2,3,5,6,4] => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,1,4,6,2,3] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,6,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,4,6,5,2,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,5,1,6,2] => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,1,4,6,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,1,6,4,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,1,6,3,5,4] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,4,2,6,1,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,3,4,6,2,5] => 0
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,3,2,6,5,4] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,5,4,6,1,2] => 0
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,3,4,1,6,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,6,3,1,5,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,1,6,4,5,3] => 0
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,3,4,6,5,2] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 0
[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
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,7,4,5] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => 0
[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
[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
[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
[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] => 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
[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
[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] => 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
[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] => 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
[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
[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
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 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
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [1,4,2,7,6,3,5] => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [1,2,6,3,7,5,4] => 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
[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
[1,1,1,0,1,0,0,1,0,1,0,0] => [6,3,7,1,2,4,5] => [1,3,2,6,4,7,5] => 0
[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] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [1,2,4,6,7,3,5] => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [1,2,3,6,7,5,4] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [2,3,4,1,5,7,6] => 0
[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
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [1,3,4,2,7,6,5] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [1,3,7,2,6,5,4] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [2,3,4,5,1,7,6] => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [1,3,4,5,7,6,2] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,8,5,6] => 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [8,1,2,6,3,4,5,7] => [1,2,3,8,4,6,5,7] => 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [7,1,2,8,3,4,5,6] => [1,2,3,7,4,8,5,6] => 0
>>> Load all 133 entries. <<<
[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [8,1,2,6,3,7,4,5] => [1,8,2,6,3,7,4,5] => 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [4,1,2,5,6,8,3,7] => [4,1,5,2,6,3,8,7] => 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,8,2,4,5,6,7] => [3,1,2,4,5,8,6,7] => 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [6,1,4,2,3,8,5,7] => [6,1,2,8,4,3,5,7] => 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [7,1,5,2,3,4,8,6] => [7,1,2,8,3,5,4,6] => 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [6,1,8,2,3,4,5,7] => [1,2,3,6,4,8,5,7] => 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [8,1,5,2,3,7,4,6] => [1,8,2,5,3,7,4,6] => 0
[1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [6,1,8,2,3,7,4,5] => [1,6,2,8,3,7,4,5] => 0
[1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [8,1,4,5,2,3,6,7] => [1,4,2,5,3,8,6,7] => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,8,7] => 0
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,8,1,3,4,7,5,6] => [2,8,1,3,4,5,7,6] => 0
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [2,6,1,5,3,7,8,4] => [6,5,2,1,7,3,8,4] => 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [2,4,1,5,6,8,3,7] => [4,2,5,1,6,3,8,7] => 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [2,4,1,8,6,7,3,5] => [2,8,4,1,6,3,7,5] => 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,3,1,2,4,5,6,7] => [1,3,2,4,5,8,6,7] => 0
[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [8,4,1,2,3,7,5,6] => [1,8,2,4,3,7,5,6] => 0
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [7,4,1,8,2,3,5,6] => [1,4,2,7,3,8,5,6] => 0
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [4,3,1,5,6,8,2,7] => [4,3,5,1,6,2,8,7] => 0
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [2,8,4,1,3,5,6,7] => [2,1,4,3,8,5,6,7] => 0
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [2,8,5,1,7,3,4,6] => [2,1,8,5,3,7,4,6] => 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [8,3,5,1,2,4,6,7] => [1,3,2,5,4,8,6,7] => 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [2,3,7,8,1,4,5,6] => [2,3,1,4,5,7,8,6] => 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => [2,3,4,5,6,1,8,7] => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2,8,4,5,6,7,1,3] => [2,1,4,5,8,6,7,3] => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,8,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,9,1] => 0
[] => [1] => [1] => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,10,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,11,1] => 0
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searching the database for the individual values of this statistic
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Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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.).