Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000664: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [1,2] => [1,2] => 0
[1,1,0,0] => [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [2,1,3] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0] => [1,3,2] => [3,2,1] => [2,3,1] => 0
[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => [3,2,1] => 0
[1,1,1,0,0,0] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [1,4,2,3] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 0
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [3,2,4,1] => [2,4,1,3] => 0
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,1,3,2] => [3,4,2,1] => 0
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [2,4,3,1] => [3,4,1,2] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [3,4,2,1] => [4,1,3,2] => 0
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [3,4,1,2] => [3,1,4,2] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [1,4,3,5,2] => [1,3,5,2,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,3,2] => 0
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [1,4,2,5,3] => [1,5,3,2,4] => 0
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [1,3,5,4,2] => [1,4,5,2,3] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [1,4,5,3,2] => [1,5,2,4,3] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => [1,4,2,5,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,1,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,1,3] => 0
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [4,2,5,3,1] => [2,5,1,4,3] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 0
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,1,3,5,2] => [3,5,2,1,4] => 0
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [5,1,2,4,3] => [4,5,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [4,1,2,5,3] => [5,3,2,1,4] => 0
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,1,5,4,2] => [4,5,2,1,3] => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [4,1,5,3,2] => [5,2,1,4,3] => 0
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [4,1,5,2,3] => [4,2,1,5,3] => 0
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,1,4,5,2] => [5,2,1,3,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,1,2] => 0
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [2,4,3,5,1] => [3,5,1,2,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [3,5,2,4,1] => [4,5,1,3,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [4,5,2,3,1] => [5,1,4,3,2] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [3,4,2,5,1] => [5,1,3,2,4] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [3,5,1,4,2] => [3,1,4,5,2] => 0
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [4,5,1,3,2] => [4,3,1,5,2] => 0
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => [5,3,1,4,2] => 0
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [3,4,1,5,2] => [3,1,5,2,4] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [2,3,5,4,1] => [4,5,1,2,3] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [2,4,5,3,1] => [5,1,2,4,3] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [3,4,5,2,1] => [4,2,5,1,3] => 0
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [3,4,5,1,2] => [5,2,4,1,3] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [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] => [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [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] => [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,4,5,6,3] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,4,6,3,5] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [1,2,6,3,5,4] => [1,2,5,6,4,3] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [1,2,6,3,4,5] => [1,2,6,5,4,3] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [1,2,5,3,6,4] => [1,2,6,4,3,5] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [1,2,4,6,5,3] => [1,2,5,6,3,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [1,2,5,6,4,3] => [1,2,6,3,5,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [1,2,5,6,3,4] => [1,2,5,3,6,4] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [1,6,3,4,5,2] => [1,3,4,5,6,2] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [1,5,3,4,6,2] => [1,3,4,6,2,5] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [1,4,3,6,5,2] => [1,3,5,6,2,4] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [1,5,3,6,4,2] => [1,3,6,2,5,4] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [1,4,3,5,6,2] => [1,3,6,2,4,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [1,6,2,4,5,3] => [1,4,5,6,3,2] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [1,5,2,4,6,3] => [1,4,6,3,2,5] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [1,6,2,3,5,4] => [1,5,6,4,3,2] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [1,5,2,3,6,4] => [1,6,4,3,2,5] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [1,4,2,6,5,3] => [1,5,6,3,2,4] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [1,5,2,6,4,3] => [1,6,3,2,5,4] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [1,5,2,6,3,4] => [1,5,3,2,6,4] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => [1,6,3,2,4,5] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [1,3,6,4,5,2] => [1,4,5,6,2,3] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [1,3,5,4,6,2] => [1,4,6,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [1,4,6,3,5,2] => [1,5,6,2,4,3] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [1,5,6,3,4,2] => [1,6,2,5,4,3] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [1,4,5,3,6,2] => [1,6,2,4,3,5] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [1,4,6,2,5,3] => [1,4,2,5,6,3] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [1,5,6,2,4,3] => [1,5,4,2,6,3] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [1,5,6,2,3,4] => [1,6,4,2,5,3] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [1,4,5,2,6,3] => [1,4,2,6,3,5] => 0
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Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
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
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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