Mp00033: Dyck paths to two-row standard tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
[1,0]=>[[1],[2]]=>[2,1]=>[2,1]=>[1,2] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[2,1,4,3]=>[1,2,3,4] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[1,3,4,2]=>[1,2,3,4] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[2,1,4,3,6,5]=>[1,2,3,4,5,6] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[2,1,3,5,6,4]=>[1,2,3,4,5,6] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[1,3,4,2,6,5]=>[1,2,3,4,5,6] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[1,3,2,5,6,4]=>[1,2,3,4,5,6] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[1,2,4,5,6,3]=>[1,2,3,4,5,6] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[2,1,4,3,6,5,8,7]=>[1,2,3,4,5,6,7,8] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>[2,1,3,4,6,7,8,5]=>[1,2,3,4,5,6,7,8] [1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[3,4,7,8,1,2,5,6]=>[1,3,4,2,5,7,8,6]=>[1,2,3,4,5,6,7,8] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[3,5,7,8,1,2,4,6]=>[1,3,2,5,4,7,8,6]=>[1,2,3,4,5,6,7,8] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[1,3,2,4,6,7,8,5]=>[1,2,3,4,5,6,7,8] [1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[4,5,7,8,1,2,3,6]=>[1,2,4,5,3,7,8,6]=>[1,2,3,4,5,6,7,8] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[1,2,4,3,6,7,8,5]=>[1,2,3,4,5,6,7,8] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[1,2,3,5,6,7,8,4]=>[1,2,3,4,5,6,7,8]
to two-row standard tableau
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
reading word permutation
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Foata bijection
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.).
cycle-as-one-line notation
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.