Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000255: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 1
[1,0,1,0,1,0] => [3,2,1] => [2,3,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 2
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,4,2,1] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,2,1] => 1
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,4,3,1] => 2
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [2,3,4,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [3,2,4,1] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [3,4,1,2] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,3,1,2] => 1
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,4,2,3] => 3
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,3,2,4] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [4,5,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [3,4,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [4,3,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [4,5,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,4,2,3,1] => 1
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,5,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [3,4,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [4,3,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,5,3,4,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [2,3,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,2,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [4,5,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [5,4,3,1,2] => 1
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,5,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [3,4,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [4,3,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [4,5,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [5,4,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,5,2,1,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,5,3,1,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [3,2,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [4,5,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,4,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [3,5,1,2,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [3,4,1,2,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,1,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,5,1,3,4] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,4,1,3,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,5,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,4,2,3,5] => 3
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [5,6,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [4,5,6,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [5,4,6,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [5,6,3,4,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [4,6,3,5,2,1] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [4,5,3,6,2,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [5,4,3,6,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [3,6,4,5,2,1] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [3,5,4,6,2,1] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [3,4,5,6,2,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [4,3,5,6,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [5,6,4,2,3,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,6,5,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [4,5,6,2,3,1] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [5,4,6,2,3,1] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [5,6,3,2,4,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [6,5,3,2,4,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [4,6,3,2,5,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [4,5,3,2,6,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [5,4,3,2,6,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [3,6,4,2,5,1] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [3,5,4,2,6,1] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [3,4,5,2,6,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [4,3,5,2,6,1] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [5,6,2,3,4,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [4,6,2,3,5,1] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [4,5,2,3,6,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [5,4,2,3,6,1] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,6,2,4,5,1] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,5,2,4,6,1] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [3,4,2,5,6,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [4,3,2,5,6,1] => 1
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Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
Map
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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