Identifier
Values
[1] => [1,0] => [1,0] => [2,1] => 0
[2] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 0
[1,1] => [1,1,0,0] => [1,0,1,0] => [3,1,2] => 1
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 1
[2,2] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 1
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 1
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
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Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.