Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [2,3,1] => 0
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [3,4,1,2] => 0
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [4,2,3,1] => 0
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,5,1,2,3] => 0
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,3,1,4] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,6,1,2,3,4] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [2,4,5,3,1] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,4,1,2] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,4,2,5,1] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,3,1,4,5] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,4,5,1,2] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,5,2,3,1] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,5,3,4,1] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [3,5,2,4,6,1] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [3,5,6,4,1,2] => 0
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,6,2,4,3,1] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,5,1,2,3] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,6,5,2,3,1] => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,4,1,2,5] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,5,1] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,5,6,1,2,3] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [2,3,5,6,4,1] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,6,2,3,1,4] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [2,6,4,5,3,1] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [4,5,3,6,1,2] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [3,4,5,2,6,1] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,6,3,4,1,5] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [2,4,5,6,3,1] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [5,6,3,4,1,2] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [3,4,2,5,6,1] => 0
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,4,5,1,2] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,5,2,3,1] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,4,5,3,6,1] => 0
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,4,5,6,1,2] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,5,6,2,3,1] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,5,6,3,4,1] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,6,4,5,1] => 0
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[] => [] => [1] => [1] => 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 number of occurrences of the Hertzsprung pattern 132 in a permutation.
A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.