Identifier
Values
[1,0] => [[1],[2]] => [2,1] => {{1,2}} => 0
[1,0,1,0] => [[1,3],[2,4]] => [2,4,1,3] => {{1,2,3,4}} => 0
[1,1,0,0] => [[1,2],[3,4]] => [3,4,1,2] => {{1,3},{2,4}} => 0
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => {{1,2,4},{3,5,6}} => 0
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => {{1,2,3,4,5,6}} => 0
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => {{1,2,3,4,5,6}} => 0
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => {{1,3,4,6},{2,5}} => 1
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}} => 0
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}} => 2
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,4,7,8,1,3,5,6] => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,6,7,8,1,3,4,5] => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [3,4,6,8,1,2,5,7] => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [3,5,7,8,1,2,4,6] => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [3,6,7,8,1,2,4,5] => {{1,3,4,5,7,8},{2,6}} => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [4,5,6,8,1,2,3,7] => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [4,5,7,8,1,2,3,6] => {{1,2,4,5,6,8},{3,7}} => 1
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}} => 0
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 rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a rcs (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.