Identifier
Values
[(1,2)] => [2,1] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => 0
[(1,4),(2,3)] => [4,3,2,1] => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => 0
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,2,5,3,6,1] => 2
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [4,2,5,1,6,3] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => 0
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [4,3,5,1,6,2] => 2
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 3
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 0
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => 0
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 1
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => 1
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => 2
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => 2
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => 3
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => 3
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 2
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => 2
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => 0
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => 0
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => 0
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => 1
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => 1
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 3
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 3
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 4
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => 4
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => 3
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => 2
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => 2
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => 1
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => 1
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => 1
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => 2
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => 2
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => 2
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => 3
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => 3
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => 4
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => 5
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => 5
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => 4
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => 3
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [5,1,6,3,7,4,8,2] => 2
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => 2
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [3,1,6,4,7,5,8,2] => 2
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,6,4,7,5,8,3] => 2
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,6,4,7,3,8,5] => 1
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [3,1,6,4,7,2,8,5] => 1
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => 1
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [5,1,6,3,7,2,8,4] => 1
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => 2
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => 3
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => 4
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => 3
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => 2
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [5,2,6,1,7,3,8,4] => 1
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 0
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,1,6,2,7,3,8,5] => 0
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,1,6,2,7,4,8,5] => 0
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,6,3,7,4,8,5] => 0
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => 0
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => 0
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,1,5,2,7,3,8,6] => 0
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [4,2,5,1,7,3,8,6] => 1
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [4,2,6,1,7,3,8,5] => 1
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [4,2,6,3,7,1,8,5] => 2
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [4,2,6,3,7,5,8,1] => 3
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [3,2,6,4,7,5,8,1] => 3
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [3,2,6,4,7,1,8,5] => 2
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [3,2,6,1,7,4,8,5] => 1
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [3,2,5,1,7,4,8,6] => 1
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [3,2,4,1,7,5,8,6] => 1
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,1,4,2,7,5,8,6] => 0
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,8,6] => 0
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,4,3,7,6,8,5] => 1
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [3,1,4,2,7,6,8,5] => 1
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => 2
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => 2
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => 2
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [3,2,6,5,7,1,8,4] => 3
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => 4
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [4,2,6,5,7,3,8,1] => 4
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [4,2,6,5,7,1,8,3] => 3
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [4,2,6,1,7,5,8,3] => 2
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => 2
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [4,1,5,2,7,6,8,3] => 1
>>> Load all 126 entries. <<<
[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,8,2,7,6,4] => [3,1,5,2,7,6,8,4] => 1
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,8,3,7,6,4] => [2,1,5,3,7,6,8,4] => 1
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [2,1,6,3,7,5,8,4] => 1
[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,8,7,2,5,4] => [3,1,6,2,7,5,8,4] => 1
[(1,4),(2,6),(3,8),(5,7)] => [4,6,8,1,7,2,5,3] => [4,1,6,2,7,5,8,3] => 1
[(1,5),(2,6),(3,8),(4,7)] => [5,6,8,7,1,2,4,3] => [5,1,6,2,7,4,8,3] => 1
[(1,6),(2,5),(3,8),(4,7)] => [6,5,8,7,2,1,4,3] => [5,2,6,1,7,4,8,3] => 2
[(1,7),(2,5),(3,8),(4,6)] => [7,5,8,6,2,4,1,3] => [5,2,6,4,7,1,8,3] => 3
[(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [5,2,6,4,7,3,8,1] => 4
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [5,4,6,2,7,3,8,1] => 5
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => 4
[(1,6),(2,7),(3,8),(4,5)] => [6,7,8,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => 3
[(1,5),(2,7),(3,8),(4,6)] => [5,7,8,6,1,4,2,3] => [5,1,6,4,7,2,8,3] => 2
[(1,4),(2,7),(3,8),(5,6)] => [4,7,8,1,6,5,2,3] => [4,1,6,5,7,2,8,3] => 2
[(1,3),(2,7),(4,8),(5,6)] => [3,7,1,8,6,5,2,4] => [3,1,6,5,7,2,8,4] => 2
[(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => [2,1,6,5,7,3,8,4] => 2
[(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 3
[(1,3),(2,8),(4,7),(5,6)] => [3,8,1,7,6,5,4,2] => [3,1,6,5,7,4,8,2] => 3
[(1,4),(2,8),(3,7),(5,6)] => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 3
[(1,5),(2,8),(3,7),(4,6)] => [5,8,7,6,1,4,3,2] => [5,1,6,4,7,3,8,2] => 3
[(1,6),(2,8),(3,7),(4,5)] => [6,8,7,5,4,1,3,2] => [5,4,6,1,7,3,8,2] => 4
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => 5
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => 6
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 0
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 0
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Description
The number of occurrences of the pattern 23-1.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $23\!\!-\!\!1$.
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
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.