Identifier
Values
[1] => [1,0] => [1] => [1] => 0
[2] => [1,0,1,0] => [2,1] => [2,1] => 0
[1,1] => [1,1,0,0] => [1,2] => [1,2] => 0
[3] => [1,0,1,0,1,0] => [2,1,3] => [3,1,2] => 0
[2,1] => [1,0,1,1,0,0] => [2,3,1] => [1,3,2] => 0
[1,1,1] => [1,1,0,1,0,0] => [1,3,2] => [2,3,1] => 0
[4] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [4,1,3,2] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [4,3,2,1] => 0
[2,2] => [1,1,1,0,0,0] => [1,2,3] => [3,2,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [1,4,2,3] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => [2,4,1,3] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [5,1,3,2,4] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [5,3,2,4,1] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,1,3] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [1,5,3,4,2] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [3,4,1,2] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [1,5,2,4,3] => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [2,5,1,4,3] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [6,1,3,2,5,4] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [6,3,2,5,1,4] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [5,3,4,2,1] => 0
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [1,6,3,5,2,4] => 1
[3,3] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => [4,3,1,2] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [1,5,2,3,4] => 0
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [1,6,3,4,2,5] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [4,2,3,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [3,5,1,4,2] => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [1,6,2,4,3,5] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => [2,6,1,4,3,5] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [6,3,5,2,4,1] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [5,2,1,3,4] => 0
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [1,6,3,4,5,2] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [3,5,1,2,4] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,1,4] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [1,6,2,5,3,4] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [4,5,1,2,3] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,3,6] => [3,6,1,4,2,5] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [6,3,1,4,5,2] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [5,3,1,2,4] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [1,6,2,3,5,4] => 0
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [6,3,4,5,2,1] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [5,3,4,1,2] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => [3,6,1,5,2,4] => 0
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,3,1,4,5,6] => [1,6,2,3,4,5] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [5,2,4,1,3] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,6,3,4] => [4,6,1,5,2,3] => 0
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [6,2,1,3,5,4] => 0
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => [3,6,1,2,5,4] => 0
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [6,2,3,5,4,1] => 1
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [5,2,3,4,1] => 1
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,5,6] => [3,6,1,2,4,5] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => [6,2,3,1,4,5] => 1
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => [4,6,1,2,3,5] => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => [6,3,1,2,5,4] => 0
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5] => [6,3,5,1,2,4] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,1,5] => 2
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,2,3,4,5] => [5,6,1,2,3,4] => 0
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => [6,3,4,1,2,5] => 1
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => [6,2,4,1,3,5] => 1
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,2,4,5,6,3] => [6,3,4,5,1,2] => 1
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,2,3,6,4,5] => [6,2,5,1,3,4] => 0
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5] => [6,2,3,5,1,4] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 2
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Description
The number of non-records in a permutation.
A record in a permutation $\pi$ is a value $\pi(j)$ which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum.
For example, in the permutation $\pi = [1, 4, 3, 2, 5]$, the values $1$ is a left-to-right minimum, $1, 4, 5$ are left-to-right maxima, $5, 2, 1$ are right-to-left minima and $5$ is a right-to-left maximum. Hence, $3$ is the unique non-record.
Permutations without non-records are called square [1].
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
toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.