Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1
[1,0]=>[[1],[2]]=>[2,1]=>[2,1] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[4,2,1,3] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[4,1,3,2] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[6,2,1,4,3,5] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[6,2,1,3,5,4] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[6,1,3,4,2,5] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[6,1,3,2,5,4] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[8,2,1,4,3,6,5,7] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>[8,2,1,3,4,6,7,5] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[8,1,3,2,4,6,7,5] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[8,1,2,4,5,6,3,7] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[8,1,2,4,3,6,7,5] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4] [1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[2,4,6,8,10,1,3,5,7,9]=>[10,2,1,4,3,6,5,8,7,9] [1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[2,4,6,8,10,12,1,3,5,7,9,11]=>[12,2,1,4,3,6,5,8,7,10,9,11]
Map
to two-row standard tableau
Description
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.
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
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
  • $des$ is the number of descents, St000021The number of descents of a permutation.,
  • $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
  • $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
  • $Ebot$ is the sum of the excedance bottoms,
  • $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
  • $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
  • $Res$ is the sum of the (right) embracing numbers,
  • $Ine$ is the sum of the side numbers.