Identifier
Mp00033:
Dyck paths
—to two-row standard tableau⟶
Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-2Cc0002;cc-rep-4
[1,0]=>[[1],[2]]=>[[1,2]]=>[1,2]=>[1,1]
[1,0,1,0]=>[[1,3],[2,4]]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,1]
[1,1,0,0]=>[[1,2],[3,4]]=>[[1,3],[2,4]]=>[2,4,1,3]=>[2,1,1]
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[3,1,1,1]
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[3,1,1,1]
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[3,1,1,1]
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[3,1,1,1]
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[3,1,1,1]
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[4,1,1,1,1]
[1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[1,2],[3,4],[5,7],[6,8]]=>[6,8,5,7,3,4,1,2]=>[4,1,1,1,1]
[1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[1,2],[3,5],[4,6],[7,8]]=>[7,8,4,6,3,5,1,2]=>[4,1,1,1,1]
[1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[[1,2],[3,5],[4,7],[6,8]]=>[6,8,4,7,3,5,1,2]=>[4,1,1,1,1]
[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[1,2],[3,6],[4,7],[5,8]]=>[5,8,4,7,3,6,1,2]=>[4,1,1,1,1]
[1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[1,3],[2,4],[5,6],[7,8]]=>[7,8,5,6,2,4,1,3]=>[4,1,1,1,1]
[1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[1,3],[2,4],[5,7],[6,8]]=>[6,8,5,7,2,4,1,3]=>[4,1,1,1,1]
[1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[[1,3],[2,5],[4,6],[7,8]]=>[7,8,4,6,2,5,1,3]=>[4,1,1,1,1]
[1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[[1,3],[2,5],[4,7],[6,8]]=>[6,8,4,7,2,5,1,3]=>[4,1,1,1,1]
[1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[[1,3],[2,6],[4,7],[5,8]]=>[5,8,4,7,2,6,1,3]=>[4,1,1,1,1]
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[1,4],[2,5],[3,6],[7,8]]=>[7,8,3,6,2,5,1,4]=>[4,1,1,1,1]
[1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[[1,4],[2,5],[3,7],[6,8]]=>[6,8,3,7,2,5,1,4]=>[4,1,1,1,1]
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[[1,4],[2,6],[3,7],[5,8]]=>[5,8,3,7,2,6,1,4]=>[4,1,1,1,1]
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[1,5],[2,6],[3,7],[4,8]]=>[4,8,3,7,2,6,1,5]=>[4,1,1,1,1]
[1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,3,4,1,2]=>[5,1,1,1,1,1]
[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>[6,1,1,1,1,1,1]
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.
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
conjugate
Description
Sends a standard tableau to its conjugate tableau.
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
LLPS
Description
The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
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