Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
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]=>[[1,2],[3]]=>[3,1,2]=>[3,1,2]=>[3,1,2]
[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]=>[[1,2,3],[4]]=>[4,1,2,3]=>[4,1,2,3]=>[4,1,2,3]
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[4,1,3,2]=>[3,4,1,2]
[2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[3,1,4,2]=>[4,3,1,2]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[2,3,4,1]=>[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]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[5,1,2,3,4]=>[5,1,2,3,4]
[3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[5,1,2,4,3]=>[4,5,1,2,3]
[3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[4,1,2,5,3]=>[5,4,1,2,3]
[2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[4,1,5,3,2]=>[5,3,4,1,2]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[3,1,4,5,2]=>[5,4,3,1,2]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[2,3,4,5,1]=>[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]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[6,1,2,3,4,5]=>[6,1,2,3,4,5]
[4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[6,1,2,3,5,4]=>[5,6,1,2,3,4]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[5,1,2,3,6,4]=>[6,5,1,2,3,4]
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[4,5,6,1,2,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[5,1,2,6,4,3]=>[6,4,5,1,2,3]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[4,1,2,5,6,3]=>[6,5,4,1,2,3]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[4,1,6,3,5,2]=>[5,6,3,4,1,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[4,1,5,3,6,2]=>[6,5,3,4,1,2]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[3,1,4,5,6,2]=>[6,5,4,3,1,2]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[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]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[7,1,2,3,4,6,5]=>[6,7,1,2,3,4,5]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[6,1,2,3,4,7,5]=>[7,6,1,2,3,4,5]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[7,1,2,3,5,6,4]=>[5,6,7,1,2,3,4]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[6,1,2,3,7,5,4]=>[7,5,6,1,2,3,4]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[5,1,2,3,6,7,4]=>[7,6,5,1,2,3,4]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[6,1,2,7,4,5,3]=>[7,4,5,6,1,2,3]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[5,1,2,7,4,6,3]=>[6,7,4,5,1,2,3]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[5,1,2,6,4,7,3]=>[7,6,4,5,1,2,3]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[4,1,2,5,6,7,3]=>[7,6,5,4,1,2,3]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[4,1,6,3,7,5,2]=>[7,5,6,3,4,1,2]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[4,1,5,3,6,7,2]=>[7,6,5,3,4,1,2]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[3,1,4,5,6,7,2]=>[7,6,5,4,3,1,2]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[2,3,4,5,6,7,1]=>[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]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,7]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[8,1,2,3,4,6,7,5]=>[6,7,8,1,2,3,4,5]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[7,1,2,3,4,8,6,5]=>[8,6,7,1,2,3,4,5]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[6,1,2,3,4,7,8,5]=>[8,7,6,1,2,3,4,5]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]=>[5,6,7,8,1,2,3,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[7,1,2,3,8,5,6,4]=>[8,5,6,7,1,2,3,4]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,7,4]=>[7,8,5,6,1,2,3,4]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[6,1,2,3,7,5,8,4]=>[8,7,5,6,1,2,3,4]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[5,1,2,3,6,7,8,4]=>[8,7,6,5,1,2,3,4]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>[6,1,2,8,4,5,7,3]=>[7,8,4,5,6,1,2,3]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[6,1,2,7,4,5,8,3]=>[8,7,4,5,6,1,2,3]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[4,1,2,5,6,7,8,3]=>[8,7,6,5,4,1,2,3]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]=>[7,8,5,6,3,4,1,2]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[4,1,6,3,7,5,8,2]=>[8,7,5,6,3,4,1,2]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[4,1,5,3,6,7,8,2]=>[8,7,6,5,3,4,1,2]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[3,1,4,5,6,7,8,2]=>[8,7,6,5,4,3,1,2]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]=>[8,7,6,5,4,3,2,1]
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
reading word permutation
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
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
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 $\chi(\pi)$.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
searching the database
Sorry, this map was not found in the database.