Identifier
Mp00113:
Perfect matchings
—reverse⟶
Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Images
=>
Cc0012;cc-rep-0Cc0012;cc-rep-1Cc0002;cc-rep-3
[(1,2)]=>[(1,2)]=>[2,1]=>[2]
[(1,2),(3,4)]=>[(1,2),(3,4)]=>[2,1,4,3]=>[2,2]
[(1,3),(2,4)]=>[(1,3),(2,4)]=>[3,4,1,2]=>[2,1,1]
[(1,4),(2,3)]=>[(1,4),(2,3)]=>[4,3,2,1]=>[4]
[(1,2),(3,4),(5,6)]=>[(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,2,2]
[(1,3),(2,4),(5,6)]=>[(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,2,1,1]
[(1,4),(2,3),(5,6)]=>[(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[4,2]
[(1,5),(2,3),(4,6)]=>[(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[4,1,1]
[(1,6),(2,3),(4,5)]=>[(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[4,2]
[(1,6),(2,4),(3,5)]=>[(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,1,1]
[(1,5),(2,4),(3,6)]=>[(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,1]
[(1,4),(2,5),(3,6)]=>[(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[2,1,1,1,1]
[(1,3),(2,5),(4,6)]=>[(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[2,2,1,1]
[(1,2),(3,5),(4,6)]=>[(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[2,2,1,1]
[(1,2),(3,6),(4,5)]=>[(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[4,2]
[(1,3),(2,6),(4,5)]=>[(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[4,1,1]
[(1,4),(2,6),(3,5)]=>[(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,1,1]
[(1,5),(2,6),(3,4)]=>[(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,1,1]
[(1,6),(2,5),(3,4)]=>[(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[6]
[(1,2),(3,4),(5,6),(7,8)]=>[(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)]=>[(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,2,2,1,1]
[(1,4),(2,3),(5,6),(7,8)]=>[(1,2),(3,4),(5,8),(6,7)]=>[2,1,4,3,8,7,6,5]=>[4,2,2]
[(1,5),(2,3),(4,6),(7,8)]=>[(1,2),(3,5),(4,8),(6,7)]=>[2,1,5,8,3,7,6,4]=>[4,2,1,1]
[(1,6),(2,3),(4,5),(7,8)]=>[(1,2),(3,8),(4,5),(6,7)]=>[2,1,8,5,4,7,6,3]=>[4,2,2]
[(1,7),(2,3),(4,5),(6,8)]=>[(1,3),(2,8),(4,5),(6,7)]=>[3,8,1,5,4,7,6,2]=>[4,2,1,1]
[(1,8),(2,3),(4,5),(6,7)]=>[(1,8),(2,3),(4,5),(6,7)]=>[8,3,2,5,4,7,6,1]=>[4,2,2]
[(1,8),(2,4),(3,5),(6,7)]=>[(1,8),(2,3),(4,6),(5,7)]=>[8,3,2,6,7,4,5,1]=>[4,2,1,1]
[(1,7),(2,4),(3,5),(6,8)]=>[(1,3),(2,8),(4,6),(5,7)]=>[3,8,1,6,7,4,5,2]=>[4,1,1,1,1]
[(1,6),(2,4),(3,5),(7,8)]=>[(1,2),(3,8),(4,6),(5,7)]=>[2,1,8,6,7,4,5,3]=>[4,2,1,1]
[(1,5),(2,4),(3,6),(7,8)]=>[(1,2),(3,6),(4,8),(5,7)]=>[2,1,6,8,7,3,5,4]=>[4,2,1,1]
[(1,4),(2,5),(3,6),(7,8)]=>[(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,2,1,1,1,1]
[(1,3),(2,5),(4,6),(7,8)]=>[(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,2,2,1,1]
[(1,2),(3,5),(4,6),(7,8)]=>[(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,2,2,1,1]
[(1,2),(3,6),(4,5),(7,8)]=>[(1,2),(3,6),(4,5),(7,8)]=>[2,1,6,5,4,3,8,7]=>[4,2,2]
[(1,3),(2,6),(4,5),(7,8)]=>[(1,2),(3,7),(4,5),(6,8)]=>[2,1,7,5,4,8,3,6]=>[4,2,1,1]
[(1,4),(2,6),(3,5),(7,8)]=>[(1,2),(3,7),(4,6),(5,8)]=>[2,1,7,6,8,4,3,5]=>[4,2,1,1]
[(1,5),(2,6),(3,4),(7,8)]=>[(1,2),(3,7),(4,8),(5,6)]=>[2,1,7,8,6,5,3,4]=>[4,2,1,1]
[(1,6),(2,5),(3,4),(7,8)]=>[(1,2),(3,8),(4,7),(5,6)]=>[2,1,8,7,6,5,4,3]=>[6,2]
[(1,7),(2,5),(3,4),(6,8)]=>[(1,3),(2,8),(4,7),(5,6)]=>[3,8,1,7,6,5,4,2]=>[6,1,1]
[(1,8),(2,5),(3,4),(6,7)]=>[(1,8),(2,3),(4,7),(5,6)]=>[8,3,2,7,6,5,4,1]=>[6,2]
[(1,8),(2,6),(3,4),(5,7)]=>[(1,8),(2,4),(3,7),(5,6)]=>[8,4,7,2,6,5,3,1]=>[6,1,1]
[(1,7),(2,6),(3,4),(5,8)]=>[(1,4),(2,8),(3,7),(5,6)]=>[4,8,7,1,6,5,3,2]=>[6,1,1]
[(1,6),(2,7),(3,4),(5,8)]=>[(1,4),(2,7),(3,8),(5,6)]=>[4,7,8,1,6,5,2,3]=>[4,1,1,1,1]
[(1,5),(2,7),(3,4),(6,8)]=>[(1,3),(2,7),(4,8),(5,6)]=>[3,7,1,8,6,5,2,4]=>[4,2,1,1]
[(1,4),(2,7),(3,5),(6,8)]=>[(1,3),(2,7),(4,6),(5,8)]=>[3,7,1,6,8,4,2,5]=>[4,1,1,1,1]
[(1,3),(2,7),(4,5),(6,8)]=>[(1,3),(2,7),(4,5),(6,8)]=>[3,7,1,5,4,8,2,6]=>[4,1,1,1,1]
[(1,2),(3,7),(4,5),(6,8)]=>[(1,3),(2,6),(4,5),(7,8)]=>[3,6,1,5,4,2,8,7]=>[4,2,1,1]
[(1,2),(3,8),(4,5),(6,7)]=>[(1,6),(2,3),(4,5),(7,8)]=>[6,3,2,5,4,1,8,7]=>[4,2,2]
[(1,3),(2,8),(4,5),(6,7)]=>[(1,7),(2,3),(4,5),(6,8)]=>[7,3,2,5,4,8,1,6]=>[4,2,1,1]
[(1,4),(2,8),(3,5),(6,7)]=>[(1,7),(2,3),(4,6),(5,8)]=>[7,3,2,6,8,4,1,5]=>[4,2,1,1]
[(1,5),(2,8),(3,4),(6,7)]=>[(1,7),(2,3),(4,8),(5,6)]=>[7,3,2,8,6,5,1,4]=>[4,3,1]
[(1,6),(2,8),(3,4),(5,7)]=>[(1,7),(2,4),(3,8),(5,6)]=>[7,4,8,2,6,5,1,3]=>[4,2,1,1]
[(1,7),(2,8),(3,4),(5,6)]=>[(1,7),(2,8),(3,4),(5,6)]=>[7,8,4,3,6,5,1,2]=>[4,2,1,1]
[(1,8),(2,7),(3,4),(5,6)]=>[(1,8),(2,7),(3,4),(5,6)]=>[8,7,4,3,6,5,2,1]=>[6,2]
[(1,8),(2,7),(3,5),(4,6)]=>[(1,8),(2,7),(3,5),(4,6)]=>[8,7,5,6,3,4,2,1]=>[6,1,1]
[(1,7),(2,8),(3,5),(4,6)]=>[(1,7),(2,8),(3,5),(4,6)]=>[7,8,5,6,3,4,1,2]=>[4,1,1,1,1]
[(1,6),(2,8),(3,5),(4,7)]=>[(1,7),(2,5),(3,8),(4,6)]=>[7,5,8,6,2,4,1,3]=>[4,2,1,1]
[(1,5),(2,8),(3,6),(4,7)]=>[(1,7),(2,5),(3,6),(4,8)]=>[7,5,6,8,2,3,1,4]=>[4,1,1,1,1]
[(1,4),(2,8),(3,6),(5,7)]=>[(1,7),(2,4),(3,6),(5,8)]=>[7,4,6,2,8,3,1,5]=>[4,2,1,1]
[(1,3),(2,8),(4,6),(5,7)]=>[(1,7),(2,4),(3,5),(6,8)]=>[7,4,5,2,3,8,1,6]=>[4,1,1,1,1]
[(1,2),(3,8),(4,6),(5,7)]=>[(1,6),(2,4),(3,5),(7,8)]=>[6,4,5,2,3,1,8,7]=>[4,2,1,1]
[(1,2),(3,7),(4,6),(5,8)]=>[(1,4),(2,6),(3,5),(7,8)]=>[4,6,5,1,3,2,8,7]=>[4,2,1,1]
[(1,3),(2,7),(4,6),(5,8)]=>[(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,1,1,1]
[(1,4),(2,7),(3,6),(5,8)]=>[(1,4),(2,7),(3,6),(5,8)]=>[4,7,6,1,8,3,2,5]=>[4,2,1,1]
[(1,5),(2,7),(3,6),(4,8)]=>[(1,5),(2,7),(3,6),(4,8)]=>[5,7,6,8,1,3,2,4]=>[4,1,1,1,1]
[(1,6),(2,7),(3,5),(4,8)]=>[(1,5),(2,7),(3,8),(4,6)]=>[5,7,8,6,1,4,2,3]=>[4,1,1,1,1]
[(1,7),(2,6),(3,5),(4,8)]=>[(1,5),(2,8),(3,7),(4,6)]=>[5,8,7,6,1,4,3,2]=>[6,1,1]
[(1,8),(2,6),(3,5),(4,7)]=>[(1,8),(2,5),(3,7),(4,6)]=>[8,5,7,6,2,4,3,1]=>[6,1,1]
[(1,8),(2,5),(3,6),(4,7)]=>[(1,8),(2,5),(3,6),(4,7)]=>[8,5,6,7,2,3,4,1]=>[4,1,1,1,1]
[(1,7),(2,5),(3,6),(4,8)]=>[(1,5),(2,8),(3,6),(4,7)]=>[5,8,6,7,1,3,4,2]=>[4,1,1,1,1]
[(1,6),(2,5),(3,7),(4,8)]=>[(1,5),(2,6),(3,8),(4,7)]=>[5,6,8,7,1,2,4,3]=>[4,1,1,1,1]
[(1,5),(2,6),(3,7),(4,8)]=>[(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[2,1,1,1,1,1,1]
[(1,4),(2,6),(3,7),(5,8)]=>[(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[2,2,1,1,1,1]
[(1,3),(2,6),(4,7),(5,8)]=>[(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[2,2,1,1,1,1]
[(1,2),(3,6),(4,7),(5,8)]=>[(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[2,2,1,1,1,1]
[(1,2),(3,5),(4,7),(6,8)]=>[(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[2,2,2,1,1]
[(1,3),(2,5),(4,7),(6,8)]=>[(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[2,2,2,1,1]
[(1,4),(2,5),(3,7),(6,8)]=>[(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[2,2,1,1,1,1]
[(1,5),(2,4),(3,7),(6,8)]=>[(1,3),(2,6),(4,8),(5,7)]=>[3,6,1,8,7,2,5,4]=>[4,2,1,1]
[(1,6),(2,4),(3,7),(5,8)]=>[(1,4),(2,6),(3,8),(5,7)]=>[4,6,8,1,7,2,5,3]=>[4,1,1,1,1]
[(1,7),(2,4),(3,6),(5,8)]=>[(1,4),(2,8),(3,6),(5,7)]=>[4,8,6,1,7,3,5,2]=>[4,2,1,1]
[(1,8),(2,4),(3,6),(5,7)]=>[(1,8),(2,4),(3,6),(5,7)]=>[8,4,6,2,7,3,5,1]=>[4,2,1,1]
[(1,8),(2,3),(4,6),(5,7)]=>[(1,8),(2,4),(3,5),(6,7)]=>[8,4,5,2,3,7,6,1]=>[4,2,1,1]
[(1,7),(2,3),(4,6),(5,8)]=>[(1,4),(2,8),(3,5),(6,7)]=>[4,8,5,1,3,7,6,2]=>[4,2,1,1]
[(1,6),(2,3),(4,7),(5,8)]=>[(1,4),(2,5),(3,8),(6,7)]=>[4,5,8,1,2,7,6,3]=>[4,1,1,1,1]
[(1,5),(2,3),(4,7),(6,8)]=>[(1,3),(2,5),(4,8),(6,7)]=>[3,5,1,8,2,7,6,4]=>[4,2,1,1]
[(1,4),(2,3),(5,7),(6,8)]=>[(1,3),(2,4),(5,8),(6,7)]=>[3,4,1,2,8,7,6,5]=>[4,2,1,1]
[(1,3),(2,4),(5,7),(6,8)]=>[(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[2,2,1,1,1,1]
[(1,2),(3,4),(5,7),(6,8)]=>[(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[2,2,2,1,1]
[(1,2),(3,4),(5,8),(6,7)]=>[(1,4),(2,3),(5,6),(7,8)]=>[4,3,2,1,6,5,8,7]=>[4,2,2]
[(1,3),(2,4),(5,8),(6,7)]=>[(1,4),(2,3),(5,7),(6,8)]=>[4,3,2,1,7,8,5,6]=>[4,2,1,1]
[(1,4),(2,3),(5,8),(6,7)]=>[(1,4),(2,3),(5,8),(6,7)]=>[4,3,2,1,8,7,6,5]=>[4,4]
[(1,5),(2,3),(4,8),(6,7)]=>[(1,5),(2,3),(4,8),(6,7)]=>[5,3,2,8,1,7,6,4]=>[4,3,1]
[(1,6),(2,3),(4,8),(5,7)]=>[(1,5),(2,4),(3,8),(6,7)]=>[5,4,8,2,1,7,6,3]=>[4,3,1]
[(1,7),(2,3),(4,8),(5,6)]=>[(1,5),(2,8),(3,4),(6,7)]=>[5,8,4,3,1,7,6,2]=>[4,3,1]
[(1,8),(2,3),(4,7),(5,6)]=>[(1,8),(2,5),(3,4),(6,7)]=>[8,5,4,3,2,7,6,1]=>[6,2]
[(1,8),(2,4),(3,7),(5,6)]=>[(1,8),(2,6),(3,4),(5,7)]=>[8,6,4,3,7,2,5,1]=>[6,1,1]
[(1,7),(2,4),(3,8),(5,6)]=>[(1,6),(2,8),(3,4),(5,7)]=>[6,8,4,3,7,1,5,2]=>[4,2,1,1]
[(1,6),(2,4),(3,8),(5,7)]=>[(1,6),(2,4),(3,8),(5,7)]=>[6,4,8,2,7,1,5,3]=>[4,2,1,1]
[(1,5),(2,4),(3,8),(6,7)]=>[(1,6),(2,3),(4,8),(5,7)]=>[6,3,2,8,7,1,5,4]=>[4,3,1]
[(1,4),(2,5),(3,8),(6,7)]=>[(1,6),(2,3),(4,7),(5,8)]=>[6,3,2,7,8,1,4,5]=>[4,1,1,1,1]
[(1,3),(2,5),(4,8),(6,7)]=>[(1,5),(2,3),(4,7),(6,8)]=>[5,3,2,7,1,8,4,6]=>[4,2,1,1]
[(1,2),(3,5),(4,8),(6,7)]=>[(1,5),(2,3),(4,6),(7,8)]=>[5,3,2,6,1,4,8,7]=>[4,2,1,1]
[(1,2),(3,6),(4,8),(5,7)]=>[(1,5),(2,4),(3,6),(7,8)]=>[5,4,6,2,1,3,8,7]=>[4,2,1,1]
[(1,3),(2,6),(4,8),(5,7)]=>[(1,5),(2,4),(3,7),(6,8)]=>[5,4,7,2,1,8,3,6]=>[4,2,1,1]
[(1,4),(2,6),(3,8),(5,7)]=>[(1,6),(2,4),(3,7),(5,8)]=>[6,4,7,2,8,1,3,5]=>[4,1,1,1,1]
[(1,5),(2,6),(3,8),(4,7)]=>[(1,6),(2,5),(3,7),(4,8)]=>[6,5,7,8,2,1,3,4]=>[4,1,1,1,1]
[(1,6),(2,5),(3,8),(4,7)]=>[(1,6),(2,5),(3,8),(4,7)]=>[6,5,8,7,2,1,4,3]=>[4,2,2]
[(1,7),(2,5),(3,8),(4,6)]=>[(1,6),(2,8),(3,5),(4,7)]=>[6,8,5,7,3,1,4,2]=>[4,2,1,1]
[(1,8),(2,5),(3,7),(4,6)]=>[(1,8),(2,6),(3,5),(4,7)]=>[8,6,5,7,3,2,4,1]=>[6,1,1]
[(1,8),(2,6),(3,7),(4,5)]=>[(1,8),(2,6),(3,7),(4,5)]=>[8,6,7,5,4,2,3,1]=>[6,1,1]
[(1,7),(2,6),(3,8),(4,5)]=>[(1,6),(2,8),(3,7),(4,5)]=>[6,8,7,5,4,1,3,2]=>[6,1,1]
[(1,6),(2,7),(3,8),(4,5)]=>[(1,6),(2,7),(3,8),(4,5)]=>[6,7,8,5,4,1,2,3]=>[4,1,1,1,1]
[(1,5),(2,7),(3,8),(4,6)]=>[(1,6),(2,7),(3,5),(4,8)]=>[6,7,5,8,3,1,2,4]=>[4,1,1,1,1]
[(1,4),(2,7),(3,8),(5,6)]=>[(1,6),(2,7),(3,4),(5,8)]=>[6,7,4,3,8,1,2,5]=>[4,1,1,1,1]
[(1,3),(2,7),(4,8),(5,6)]=>[(1,5),(2,7),(3,4),(6,8)]=>[5,7,4,3,1,8,2,6]=>[4,2,1,1]
[(1,2),(3,7),(4,8),(5,6)]=>[(1,5),(2,6),(3,4),(7,8)]=>[5,6,4,3,1,2,8,7]=>[4,2,1,1]
[(1,2),(3,8),(4,7),(5,6)]=>[(1,6),(2,5),(3,4),(7,8)]=>[6,5,4,3,2,1,8,7]=>[6,2]
[(1,3),(2,8),(4,7),(5,6)]=>[(1,7),(2,5),(3,4),(6,8)]=>[7,5,4,3,2,8,1,6]=>[6,1,1]
[(1,4),(2,8),(3,7),(5,6)]=>[(1,7),(2,6),(3,4),(5,8)]=>[7,6,4,3,8,2,1,5]=>[6,1,1]
[(1,5),(2,8),(3,7),(4,6)]=>[(1,7),(2,6),(3,5),(4,8)]=>[7,6,5,8,3,2,1,4]=>[6,1,1]
[(1,6),(2,8),(3,7),(4,5)]=>[(1,7),(2,6),(3,8),(4,5)]=>[7,6,8,5,4,2,1,3]=>[6,1,1]
[(1,7),(2,8),(3,6),(4,5)]=>[(1,7),(2,8),(3,6),(4,5)]=>[7,8,6,5,4,3,1,2]=>[6,1,1]
[(1,8),(2,7),(3,6),(4,5)]=>[(1,8),(2,7),(3,6),(4,5)]=>[8,7,6,5,4,3,2,1]=>[8]
[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[(1,2),(3,4),(5,6),(7,8),(9,10)]=>[2,1,4,3,6,5,8,7,10,9]=>[2,2,2,2,2]
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,2,2,2,2,2]
Map
reverse
Description
The reverse of a perfect matching of $\{1,2,...,n\}$.
This is the perfect matching obtained by replacing $i$ by $n+1-i$.
This is the perfect matching obtained by replacing $i$ by $n+1-i$.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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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