Identifier
Values
[1,2] => [1,2] => [1,0,1,0] => [1,1,0,1,0,0] => 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 3
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 2
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => 3
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 2
search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.