Identifier
-
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001461: Permutations ⟶ ℤ
Values
[(1,2)] => [2,1] => [2,1] => [1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 2
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => [3,1,2] => 1
[(1,4),(2,3)] => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,5] => 3
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => [3,1,4,2,5] => 2
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,1,3,2,6,5] => [4,1,3,2,5] => 3
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,1,3,2,6,4] => [5,1,3,2,4] => 2
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,1,3,2,5,4] => [1,3,2,5,4] => 3
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,1,4,2,5,3] => [1,4,2,5,3] => 2
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,1,4,2,6,3] => [5,1,4,2,3] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,3] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [3,1,5,2,4] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [2,1,5,3,4] => 2
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,3,5,4] => [2,1,3,5,4] => 3
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,1,6,2,5,4] => [3,1,2,5,4] => 2
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,1,6,2,5,3] => [4,1,2,5,3] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,2,5,1,4,3] => [2,5,1,4,3] => 2
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,2,6,1,4,3] => [5,2,1,4,3] => 3
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => 4
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8,1,4,2,5,3,7,6] => [1,4,2,5,3,7,6] => 3
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,1,4,3,7,2,6,5] => [1,4,3,7,2,6,5] => 4
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8,1,4,2,6,3,7,5] => [1,4,2,6,3,7,5] => 2
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8,1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => 3
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Description
The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
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
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
This map is undefined for the empty permutation.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
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