Identifier
Values
[(1,2)] => [2,1] => [1,1,0,0] => [1,0,1,0] => 1
[(1,2),(3,4)] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 2
[(1,3),(2,4)] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2
[(1,4),(2,3)] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 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] => [1,0,1,1,1,0,0,1,0,1,0,0] => 2
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 2
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[(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] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0] => 3
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => 3
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0] => 3
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the bounce path of $D$.
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
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.