Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000809: Permutations ⟶ ℤ
Values
[1,0,1,0] => [3,1,2] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,3,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,2] => [3,2,1] => 1
[1,1,0,0,1,0] => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [4,3,1,2] => [3,1,2] => [3,2,1] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,2,3] => [4,3,2,1] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,4,2] => [4,2,1,3] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,4,1,3] => [4,3,1,2] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,1,2] => [4,2,3,1] => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,4,1,3] => [4,3,1,2] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,4] => [5,4,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,2,3,5] => [4,3,2,1,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,3] => [5,3,2,1,4] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,4] => [5,4,2,1,3] => 4
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,4,2,3] => [5,3,4,2,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,4,2,5] => [4,2,1,3,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,5,2,4] => [5,4,2,1,3] => 4
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,2] => [5,2,1,3,4] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,5,1,3,4] => [5,4,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,3,5] => [4,3,1,2,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,4,1,5,3] => [5,3,1,2,4] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,2,4] => [5,4,2,3,1] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,3,5] => [4,3,2,1,5] => 2
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,1,2,3,4] => [5,4,3,2,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,2,3] => [4,2,5,3,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,2,5] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,4] => [5,4,2,1,3] => 4
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,1,5,2,3] => [4,2,1,5,3] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,1,5,2] => [5,2,3,1,4] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,3,5,1,4] => [5,4,1,2,3] => 5
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,4,1,3,5] => [4,3,1,2,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,5,1,3,4] => [5,4,3,1,2] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,5,4,1,3] => [5,3,4,1,2] => 4
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,5,1,2,4] => [3,1,5,4,2] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,4,1,2,3] => [4,2,5,3,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,3,4,1,2] => [5,2,3,4,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,3,4,1,5] => [4,1,2,3,5] => 3
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,3,5,1,4] => [5,4,1,2,3] => 5
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,4,5,1,3] => [4,1,2,5,3] => 4
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,4,5,1,2] => [5,2,4,1,3] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [5,1,2,3,4,6] => [5,4,3,2,1,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,2,3,6,4] => [6,4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [4,1,2,3,5,6] => [4,3,2,1,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,5,4,3] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [6,1,2,5,3,4] => [6,4,5,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [4,1,2,5,3,6] => [5,3,2,1,4,6] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [1,2,5,6,3,4] => [1,2,5,3,6,4] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => 5
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [3,1,5,2,4,6] => [5,4,2,1,3,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [3,1,2,6,4,5] => [3,2,1,6,5,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,1,5,2,6,4] => [6,4,2,1,3,5] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,3,2,5,6] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [6,1,4,2,3,5] => [6,5,3,4,2,1] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,5,2,3,4,6] => [1,5,4,3,2,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [5,1,4,2,3,6] => [5,3,4,2,1,6] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,6,5,3,2,4] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,5,3,2,6,4] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [5,1,4,2,6,3] => [6,3,4,2,1,5] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [3,1,4,2,5,6] => [4,2,1,3,5,6] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [3,1,4,6,2,5] => [6,5,2,1,3,4] => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [3,1,5,2,4,6] => [5,4,2,1,3,6] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [3,1,6,5,2,4] => [6,4,5,2,1,3] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [1,4,5,2,3,6] => [1,4,2,5,3,6] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [1,4,6,2,3,5] => [1,4,2,6,5,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,6,5,2,3,4] => [1,5,3,6,4,2] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [6,1,4,5,2,3] => [6,3,4,5,2,1] => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [3,1,4,5,2,6] => [5,2,1,3,4,6] => 3
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Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth St000029The depth of a permutation. minus the usual length St000018The number of inversions of a permutation..
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth St000029The depth of a permutation. minus the usual length St000018The number of inversions of a 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.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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.
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