Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
[(1,2)] => [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 101
[(1,3),(2,4)] => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 101
[(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 101
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 10101
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 10101
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => [3,2,4,1,6,5] => 10101
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 10101
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [6,3,2,5,4,1] => [3,2,5,4,6,1] => 10101
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [4,2,5,3,6,1] => 10101
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,4,6,2,1,3] => [4,2,5,1,6,3] => 10101
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 10101
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 10101
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 10101
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 10101
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 10101
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 10101
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,6,4,3,1,2] => [4,3,5,1,6,2] => 10101
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 10101
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => 1010101
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => 1010101
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => 1010101
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => 1010101
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => 1010101
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => 1010101
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => 1010101
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => 1010101
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => 1010101
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => 1010101
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 1010101
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 1010101
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 1010101
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => 1010101
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => 1010101
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => 1010101
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => 1010101
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => 1010101
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => 1010101
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => 1010101
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => 1010101
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => 1010101
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => 1010101
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => 1010101
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 1010101
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 1010101
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => 1010101
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => 1010101
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => 1010101
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => [5,2,6,4,7,3,8,1] => 1010101
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [5,2,6,4,7,1,8,3] => 1010101
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [6,5,8,7,2,1,4,3] => [5,2,6,1,7,4,8,3] => 1010101
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 1010101
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => 1010101
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => 1010101
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => 1010101
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => 1010101
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => 1010101
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => 1010101
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => 1010101
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => 1010101
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => 1010101
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 1010101
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 1010101
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => 1010101
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => 1010101
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => 1010101
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => 1010101
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => 1010101
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => 1010101
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => 1010101
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => 1010101
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => 1010101
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 1010101
>>> Load all 127 entries. <<<Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
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
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
searching the database
Sorry, this map was not found in the database.