Identifier
Values
[1,0] => [(1,2)] => [2,1] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 2
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => 3
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 4
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => 4
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => 6
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
click to show known generating functions       
Description
Lusztig's a-function for the symmetric group.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
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.