Identifier
-
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤ
Values
[(1,2)] => [2,1] => [2] => [[1,2]] => 2
[(1,2),(3,4)] => [2,1,4,3] => [2,2] => [[1,2],[3,4]] => 2
[(1,3),(2,4)] => [3,4,1,2] => [2,1,1] => [[1,2],[3],[4]] => 2
[(1,4),(2,3)] => [3,4,2,1] => [3,1] => [[1,2,3],[4]] => 3
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,2,2] => [[1,2],[3,4],[5,6]] => 2
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 2
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [3,2,1] => [[1,2,3],[4,5],[6]] => 3
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [3,2,1] => [[1,2,3],[4,5],[6]] => 3
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 2
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => 2
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [3,2,1] => [[1,2,3],[4,5],[6]] => 3
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,2,1] => [[1,2,3],[4,5],[6]] => 3
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => 3
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,1,1] => [[1,2,3,4],[5],[6]] => 4
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 2
[(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],[5,6],[7],[8]] => 2
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 2
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 2
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 2
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 4
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 4
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 4
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 4
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 4
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 3
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 4
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 4
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 4
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 4
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => 2
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 2
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 2
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 2
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 2
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 2
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 2
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 3
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(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],[6],[7],[8]] => 2
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 2
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 3
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 3
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 3
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 4
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 4
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 3
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 3
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Description
The last entry in the first row of a standard tableau.
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.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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