Identifier
Values
[(1,2)] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,4,1,3] => 1
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => 1
[(1,4),(2,3)] => [4,3,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,6,2,5] => 1
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 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] => [3,6,2,5,4,1] => 1
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,2,6,5,3,1] => 1
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,4,2,1,6,3] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,3,1,6,2,4] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [5,6,2,1,3,4] => 0
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 0
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [6,3,5,1,4,2] => 1
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,6,1,5,3,2] => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,1,6,4,3,2] => 0
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 1
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 1
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => 1
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [4,3,6,2,8,7,5,1] => 2
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [4,3,7,6,2,1,8,5] => 2
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [6,4,3,1,7,2,8,5] => 1
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [4,3,7,1,8,6,5,2] => 1
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [5,3,8,7,6,4,2,1] => 2
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 0
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [1,8,6,5,3,7,4,2] => 2
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [6,1,7,5,3,2,8,4] => 2
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,8,4] => 1
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [6,5,3,2,8,7,4,1] => 1
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [5,2,6,3,8,7,4,1] => 1
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [5,2,7,6,3,1,8,4] => 1
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [6,5,2,1,7,3,8,4] => 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] => 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
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => 1
[(1,7),(2,5),(3,8),(4,6)] => [7,5,8,6,2,4,1,3] => [5,2,7,1,8,6,4,3] => 0
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [6,2,8,7,5,4,3,1] => 0
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [7,6,2,1,8,5,4,3] => 0
[(1,6),(2,7),(3,8),(4,5)] => [6,7,8,5,4,1,2,3] => [6,1,7,2,8,5,4,3] => 0
[(1,3),(2,8),(4,7),(5,6)] => [3,8,1,7,6,5,4,2] => [8,7,6,3,5,1,4,2] => 0
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => [7,1,8,6,5,4,3,2] => 0
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.