Identifier
Values
[1,0] => [1] => [1] => [1,0] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,0,1,0] => 0
[1,1,0,0] => [2,1] => [2,1] => [1,1,0,0] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0] => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0] => 1
[1,1,1,0,0,0] => [3,2,1] => [2,3,1] => [1,1,0,1,0,0] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => 2
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 3
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => 3
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 4
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Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Map
to 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).
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.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
  • the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
  • the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
  • the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
  • the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
  • the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.