- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001084: Permutations ⟶ ℤ

Values

[1] => [1,0] => [2,1] => [2,1] => 0
[2] => [1,0,1,0] => [3,1,2] => [3,1,2] => 0
[1,1] => [1,1,0,0] => [2,3,1] => [3,2,1] => 0
[3] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 0
[1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => 0
[4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,2,3] => 0
[2,2] => [1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,4,2,1,3] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,5,3] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,6,3,4] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,1,5,3,2,4] => 0
[2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,5,2,3,1,4] => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,2,3,5,4] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,7,4,5] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 0
[3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,1,5,3,2] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,5,1,2,6,4] => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,3,6,4] => 0
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,8,5,6] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,5,2,1,4,3] => 0
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,1,6,2,3,4] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,6,1,4,5,2] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,1,6,4] => 0
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,8,6] => 0
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 0
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,2,5,6,3] => 0
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,5,3,1,2] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,1,6,3,4,2] => 0
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 0
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 0
[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,10,8] => 0
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [11,1,2,3,4,5,6,7,8,9,10] => 0
[] => [] => [1] => [1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3$

$F_{4} = 5$

$F_{5} = 7$

Description

The number of occurrences of the vincular pattern |1-23 in a permutation.
This is the number of occurrences of the pattern

$123$ , where the first two matched entries are the first two entries of the permutation.
In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.

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.
This map is similar to Mp00235descent views to invisible inversion bottoms, but different beginning with permutations of six elements.
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 maximima 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 deficiency positions of $\chi(\pi)$ , and
the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$ .

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

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.