Identifier
Values
[1,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,2,1] => [3,2,1] => 2
[1,1,0,0] => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,3,1,2] => [1,4,3,2] => 3
[1,1,0,1,0,0] => [4,3,1,2] => [4,2,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,3,2,1,4] => [3,5,2,1,4] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,4,2,1,3] => [2,5,4,1,3] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,3,4,2,1] => [3,5,4,2,1] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,1,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,4,3,1,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,1,2,4] => [1,3,5,2,4] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,4,2,3,1] => [2,5,4,3,1] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,2,5,3,1] => [4,5,2,3,1] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,3,1,4] => [2,3,5,1,4] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,3,4,1,2] => [1,3,5,4,2] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,4,3,2,1,5] => [4,6,3,2,1,5] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => [3,6,5,2,1,4] => 4
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => [2,6,5,4,1,3] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => [5,6,3,4,2,1] => 3
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,1,5] => [3,4,6,2,1,5] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,5,2,1,3,4] => [2,1,6,5,3,4] => 4
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [6,4,5,2,1,3] => [2,4,6,5,1,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,3,4,5,2,1] => [3,4,6,5,2,1] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,1,3,4,5] => [2,1,3,6,4,5] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 5
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,4,6,3,2,5] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,3,6,5,2,4] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,4,6,5,3,2] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,5,4,2,3,1] => [2,6,5,4,3,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => [2,4,6,3,1,5] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,2,6,5,3,1] => [6,4,5,2,3,1] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,2,6,3,1,5] => [4,2,6,3,1,5] => 4
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [6,5,2,3,1,4] => [2,3,6,5,1,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,4,5,2,3,1] => [2,4,6,5,3,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,2,4,6,3,1] => [5,2,4,6,3,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,2,3,1,4,5] => [2,3,1,6,4,5] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 4
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => [1,2,4,6,3,5] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,5,3,4,1,2] => [1,3,6,5,4,2] => 4
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,3,6,4,1,2] => [1,5,6,3,4,2] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [6,3,4,1,2,5] => [1,3,4,6,2,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,5,2,3,4,1] => [2,3,6,5,4,1] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,2,3,6,4,1] => [5,2,6,3,4,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,2,6,3,4,1] => [2,5,6,3,4,1] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,2,3,4,1,5] => [2,3,4,6,1,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 3
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [6,4,5,1,2,3] => [1,2,4,6,5,3] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,3,4,5,1,2] => [1,3,4,6,5,2] => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => [1,4,3,7,2,5,6] => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => [1,3,7,6,5,2,4] => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 2
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => [1,4,5,7,3,2,6] => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => [1,3,2,7,6,4,5] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [7,5,6,3,1,2,4] => [1,3,5,7,6,2,4] => 3
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [7,4,5,6,3,1,2] => [1,4,5,7,6,3,2] => 4
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [7,3,1,2,4,5,6] => [1,3,2,4,7,5,6] => 3
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 5
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [7,5,4,1,2,3,6] => [1,2,5,7,4,3,6] => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [7,6,4,1,2,3,5] => [1,2,4,7,6,3,5] => 3
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [7,5,6,4,1,2,3] => [1,2,5,7,6,4,3] => 4
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [7,4,1,2,3,5,6] => [1,2,4,3,7,5,6] => 3
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [7,6,5,3,4,1,2] => [1,3,7,6,5,4,2] => 5
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [7,5,3,4,1,2,6] => [1,3,5,7,4,2,6] => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [7,6,3,4,1,2,5] => [1,3,4,7,6,2,5] => 3
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [7,5,6,3,4,1,2] => [1,3,5,7,6,4,2] => 4
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [7,3,4,1,2,5,6] => [1,3,4,2,7,5,6] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 4
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [7,5,1,2,3,4,6] => [1,2,3,5,7,4,6] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => 4
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [6,4,7,5,1,2,3] => [1,2,6,7,4,5,3] => 3
[1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [7,4,5,1,2,3,6] => [1,2,4,5,7,3,6] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [7,6,3,4,5,1,2] => [1,3,4,7,6,5,2] => 4
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [6,3,7,4,5,1,2] => [1,3,6,7,4,5,2] => 3
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [7,3,4,5,1,2,6] => [1,3,4,5,7,2,6] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 3
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [7,5,6,1,2,3,4] => [1,2,3,5,7,6,4] => 3
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [7,4,5,6,1,2,3] => [1,2,4,5,7,6,3] => 3
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [7,3,4,5,6,1,2] => [1,3,4,5,7,6,2] => 3
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 2
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Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
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
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.