Identifier
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Images
=>
Cc0012;cc-rep-0
[(1,2)]=>[2,1]=>[2,1]=>[2,1] [(1,2),(3,4)]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3] [(1,3),(2,4)]=>[3,4,1,2]=>[3,1,4,2]=>[2,1,4,3] [(1,4),(2,3)]=>[4,3,2,1]=>[3,2,4,1]=>[2,1,4,3] [(1,2),(3,4),(5,6)]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [(1,3),(2,4),(5,6)]=>[3,4,1,2,6,5]=>[3,1,4,2,6,5]=>[2,1,4,3,6,5] [(1,4),(2,3),(5,6)]=>[4,3,2,1,6,5]=>[3,2,4,1,6,5]=>[2,1,4,3,6,5] [(1,5),(2,3),(4,6)]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[2,1,4,3,6,5] [(1,6),(2,3),(4,5)]=>[6,3,2,5,4,1]=>[3,2,5,4,6,1]=>[2,1,4,3,6,5] [(1,6),(2,4),(3,5)]=>[6,4,5,2,3,1]=>[4,2,5,3,6,1]=>[2,1,4,3,6,5] [(1,5),(2,4),(3,6)]=>[5,4,6,2,1,3]=>[4,2,5,1,6,3]=>[2,1,4,3,6,5] [(1,4),(2,5),(3,6)]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[2,1,4,3,6,5] [(1,3),(2,5),(4,6)]=>[3,5,1,6,2,4]=>[3,1,5,2,6,4]=>[2,1,4,3,6,5] [(1,2),(3,5),(4,6)]=>[2,1,5,6,3,4]=>[2,1,5,3,6,4]=>[2,1,4,3,6,5] [(1,2),(3,6),(4,5)]=>[2,1,6,5,4,3]=>[2,1,5,4,6,3]=>[2,1,4,3,6,5] [(1,3),(2,6),(4,5)]=>[3,6,1,5,4,2]=>[3,1,5,4,6,2]=>[2,1,4,3,6,5] [(1,4),(2,6),(3,5)]=>[4,6,5,1,3,2]=>[4,1,5,3,6,2]=>[2,1,4,3,6,5] [(1,5),(2,6),(3,4)]=>[5,6,4,3,1,2]=>[4,3,5,1,6,2]=>[2,1,4,3,6,5] [(1,6),(2,5),(3,4)]=>[6,5,4,3,2,1]=>[4,3,5,2,6,1]=>[2,1,4,3,6,5] [(1,2),(3,4),(5,6),(7,8)]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,8,7] [(1,3),(2,4),(5,6),(7,8)]=>[3,4,1,2,6,5,8,7]=>[3,1,4,2,6,5,8,7]=>[2,1,4,3,6,5,8,7] [(1,4),(2,5),(3,6),(7,8)]=>[4,5,6,1,2,3,8,7]=>[4,1,5,2,6,3,8,7]=>[2,1,4,3,6,5,8,7] [(1,3),(2,5),(4,6),(7,8)]=>[3,5,1,6,2,4,8,7]=>[3,1,5,2,6,4,8,7]=>[2,1,4,3,6,5,8,7] [(1,2),(3,5),(4,6),(7,8)]=>[2,1,5,6,3,4,8,7]=>[2,1,5,3,6,4,8,7]=>[2,1,4,3,6,5,8,7] [(1,4),(2,7),(3,5),(6,8)]=>[4,7,5,1,3,8,2,6]=>[4,1,5,3,7,2,8,6]=>[2,1,4,3,6,5,8,7] [(1,5),(2,6),(3,7),(4,8)]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[2,1,4,3,6,5,8,7] [(1,4),(2,6),(3,7),(5,8)]=>[4,6,7,1,8,2,3,5]=>[4,1,6,2,7,3,8,5]=>[2,1,4,3,6,5,8,7] [(1,3),(2,6),(4,7),(5,8)]=>[3,6,1,7,8,2,4,5]=>[3,1,6,2,7,4,8,5]=>[2,1,4,3,6,5,8,7] [(1,2),(3,6),(4,7),(5,8)]=>[2,1,6,7,8,3,4,5]=>[2,1,6,3,7,4,8,5]=>[2,1,4,3,6,5,8,7] [(1,2),(3,5),(4,7),(6,8)]=>[2,1,5,7,3,8,4,6]=>[2,1,5,3,7,4,8,6]=>[2,1,4,3,6,5,8,7] [(1,3),(2,5),(4,7),(6,8)]=>[3,5,1,7,2,8,4,6]=>[3,1,5,2,7,4,8,6]=>[2,1,4,3,6,5,8,7] [(1,4),(2,5),(3,7),(6,8)]=>[4,5,7,1,2,8,3,6]=>[4,1,5,2,7,3,8,6]=>[2,1,4,3,6,5,8,7] [(1,3),(2,4),(5,7),(6,8)]=>[3,4,1,2,7,8,5,6]=>[3,1,4,2,7,5,8,6]=>[2,1,4,3,6,5,8,7] [(1,2),(3,4),(5,7),(6,8)]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,8,6]=>[2,1,4,3,6,5,8,7]
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..