Identifier
Values
[(1,2)] => [2,1] => [1,1,0,0] => [1,2] => 0
[(1,2),(3,4)] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 1
[(1,3),(2,4)] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 3
[(1,4),(2,3)] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => 3
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 2
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,1,2,5,3,6] => 4
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,1,2,5,3,6] => 4
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 6
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 6
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 8
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 8
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 8
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 6
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,6,2,4,5] => 4
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,6,2,4,5] => 4
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => 6
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 8
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 8
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => 8
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6,8] => 3
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [6,7,8,1,2,3,4,5] => 15
[(1,10),(2,9),(3,8),(4,6),(5,7)] => [6,7,8,9,10,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,6),(2,10),(3,9),(4,7),(5,8)] => [6,7,8,9,10,1,4,5,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,6),(2,9),(3,10),(4,7),(5,8)] => [6,7,8,9,10,1,4,5,2,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,10),(2,8),(3,9),(4,6),(5,7)] => [6,7,8,9,10,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,9),(2,8),(3,7),(4,6),(5,10)] => [6,7,8,9,10,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,10),(2,9),(3,6),(4,7),(5,8)] => [6,7,8,9,10,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,9),(2,10),(3,6),(4,7),(5,8)] => [6,7,8,9,10,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,6),(2,10),(3,7),(4,8),(5,9)] => [6,7,8,9,10,1,3,4,5,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,8),(2,9),(3,6),(4,7),(5,10)] => [6,7,8,9,10,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,9),(2,8),(3,6),(4,7),(5,10)] => [6,7,8,9,10,3,4,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,9),(2,6),(3,7),(4,8),(5,10)] => [6,7,8,9,10,2,3,4,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,10),(2,6),(3,7),(4,8),(5,9)] => [6,7,8,9,10,2,3,4,5,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,6),(2,10),(3,9),(4,8),(5,7)] => [6,7,8,9,10,1,5,4,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [7,8,9,10,1,2,3,4,5,6] => 24
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Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.