Identifier
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]=>[1]=>[1] [2]=>[[1,2]]=>[1,2]=>[1,2]=>[1,2] [1,1]=>[[1],[2]]=>[2,1]=>[2,1]=>[2,1] [3]=>[[1,2,3]]=>[1,2,3]=>[1,2,3]=>[1,2,3] [2,1]=>[[1,3],[2]]=>[2,1,3]=>[2,1,3]=>[2,1,3] [1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[2,3,1]=>[3,2,1] [4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [3,1]=>[[1,3,4],[2]]=>[2,1,3,4]=>[2,1,3,4]=>[2,1,3,4] [2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[3,1,4,2]=>[3,4,1,2] [2,1,1]=>[[1,4],[2],[3]]=>[3,2,1,4]=>[2,3,1,4]=>[3,2,1,4] [1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[3,2,4,1]=>[4,3,2,1] [5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[1,2,3,4,5] [4,1]=>[[1,3,4,5],[2]]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[2,1,3,4,5] [3,2]=>[[1,2,5],[3,4]]=>[3,4,1,2,5]=>[3,1,4,2,5]=>[3,4,1,2,5] [3,1,1]=>[[1,4,5],[2],[3]]=>[3,2,1,4,5]=>[2,3,1,4,5]=>[3,2,1,4,5] [2,2,1]=>[[1,3],[2,5],[4]]=>[4,2,5,1,3]=>[2,4,1,5,3]=>[4,2,5,1,3] [2,1,1,1]=>[[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>[3,2,4,1,5]=>[4,3,2,1,5] [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[3,4,2,5,1]=>[5,4,3,2,1] [6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6] [5,1]=>[[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6] [4,2]=>[[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>[3,1,4,2,5,6]=>[3,4,1,2,5,6] [4,1,1]=>[[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>[2,3,1,4,5,6]=>[3,2,1,4,5,6] [3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[4,5,6,1,2,3] [3,2,1]=>[[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>[2,4,1,5,3,6]=>[4,2,5,1,3,6] [3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>[3,2,4,1,5,6]=>[4,3,2,1,5,6] [2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[3,4,5,1,6,2]=>[5,6,3,4,1,2] [2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[5,3,2,6,1,4] [2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>[3,4,2,5,1,6]=>[5,4,3,2,1,6] [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[6,5,4,3,2,1] [7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7] [6,1]=>[[1,3,4,5,6,7],[2]]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7] [5,2]=>[[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>[3,1,4,2,5,6,7]=>[3,4,1,2,5,6,7] [5,1,1]=>[[1,4,5,6,7],[2],[3]]=>[3,2,1,4,5,6,7]=>[2,3,1,4,5,6,7]=>[3,2,1,4,5,6,7] [4,3]=>[[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>[4,1,5,2,6,3,7]=>[4,5,6,1,2,3,7] [4,2,1]=>[[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>[2,4,1,5,3,6,7]=>[4,2,5,1,3,6,7] [4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>[4,3,2,1,5,6,7]=>[3,2,4,1,5,6,7]=>[4,3,2,1,5,6,7] [3,3,1]=>[[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>[2,5,1,6,3,7,4]=>[5,2,6,7,1,3,4] [3,2,2]=>[[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>[3,4,5,1,6,2,7]=>[5,6,3,4,1,2,7] [3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>[3,2,5,1,6,4,7]=>[5,3,2,6,1,4,7] [3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7]=>[3,4,2,5,1,6,7]=>[5,4,3,2,1,6,7] [2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>[4,2,5,6,1,7,3]=>[6,4,7,2,5,1,3] [2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>[3,4,2,6,1,7,5]=>[6,4,3,2,7,1,5] [2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7]=>[4,3,5,2,6,1,7]=>[6,5,4,3,2,1,7] [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[4,5,3,6,2,7,1]=>[7,6,5,4,3,2,1] [8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8] [7,1]=>[[1,3,4,5,6,7,8],[2]]=>[2,1,3,4,5,6,7,8]=>[2,1,3,4,5,6,7,8]=>[2,1,3,4,5,6,7,8] [6,2]=>[[1,2,5,6,7,8],[3,4]]=>[3,4,1,2,5,6,7,8]=>[3,1,4,2,5,6,7,8]=>[3,4,1,2,5,6,7,8] [6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>[3,2,1,4,5,6,7,8]=>[2,3,1,4,5,6,7,8]=>[3,2,1,4,5,6,7,8] [5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>[4,3,2,1,5,6,7,8]=>[3,2,4,1,5,6,7,8]=>[4,3,2,1,5,6,7,8] [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[5,6,7,8,1,2,3,4] [4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>[5,2,6,7,1,3,4,8]=>[2,5,1,6,3,7,4,8]=>[5,2,6,7,1,3,4,8] [4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>[5,6,3,4,1,2,7,8]=>[3,4,5,1,6,2,7,8]=>[5,6,3,4,1,2,7,8] [4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>[5,3,2,6,1,4,7,8]=>[3,2,5,1,6,4,7,8]=>[5,3,2,6,1,4,7,8] [4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8]=>[3,4,2,5,1,6,7,8]=>[5,4,3,2,1,6,7,8] [3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>[6,7,3,4,8,1,2,5]=>[3,4,6,1,7,2,8,5]=>[6,7,3,4,8,1,2,5] [3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,7,4,8,5]=>[6,3,2,7,8,1,4,5] [3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3,8]=>[4,2,5,6,1,7,3,8]=>[6,4,7,2,5,1,3,8] [3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5,8]=>[3,4,2,6,1,7,5,8]=>[6,4,3,2,7,1,5,8] [3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8]=>[4,3,5,2,6,1,7,8]=>[6,5,4,3,2,1,7,8] [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[5,3,6,4,7,1,8,2]=>[7,8,5,6,3,4,1,2] [2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>[7,5,3,8,2,6,1,4]=>[3,5,2,6,7,1,8,4]=>[7,5,3,8,2,6,1,4] [2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6]=>[4,3,5,2,7,1,8,6]=>[7,5,4,3,2,8,1,6] [2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8]=>[4,5,3,6,2,7,1,8]=>[7,6,5,4,3,2,1,8] [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[5,4,6,3,7,2,8,1]=>[8,7,6,5,4,3,2,1] [9]=>[[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9] [8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>[2,1,3,4,5,6,7,8,9]=>[2,1,3,4,5,6,7,8,9]=>[2,1,3,4,5,6,7,8,9] [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9,10] [9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>[2,1,3,4,5,6,7,8,9,10]=>[2,1,3,4,5,6,7,8,9,10]=>[2,1,3,4,5,6,7,8,9,10] [1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,2,1]=>[6,5,7,4,8,3,9,2,10,1]=>[10,9,8,7,6,5,4,3,2,1]
Map
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
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
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
• the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
• the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
• the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
• the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
• the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.