Identifier
Values
[1] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 1
[2] => [1,1,0,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => 0
[1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 0
[2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 1
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 1
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 1
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 1
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 1
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 1
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 1
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 1
[4,3,2,1] => [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] => 1
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Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.