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Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001461: Permutations ⟶ ℤ

Values

[1,0] => [2,1] => {{1,2}} => [2,1] => 1
[1,0,1,0] => [3,1,2] => {{1,3},{2}} => [3,2,1] => 2
[1,1,0,0] => [2,3,1] => {{1,2,3}} => [2,3,1] => 1
[1,0,1,0,1,0] => [4,1,2,3] => {{1,4},{2},{3}} => [4,2,3,1] => 3
[1,0,1,1,0,0] => [3,1,4,2] => {{1,3,4},{2}} => [3,2,4,1] => 2
[1,1,0,0,1,0] => [2,4,1,3] => {{1,2,4},{3}} => [2,4,3,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => {{1,4},{2,3}} => [4,3,2,1] => 2
[1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => {{1,3,4,5},{2}} => [3,2,4,5,1] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 3
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => {{1,2,4,5},{3}} => [2,4,3,5,1] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => {{1,4,5},{2,3}} => [4,3,2,5,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => {{1,2,3,5},{4}} => [2,3,5,4,1] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => {{1,2,5},{3,4}} => [2,5,4,3,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => {{1,5},{2,3,4}} => [5,3,4,2,1] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => 4
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}} => [6,2,4,5,3,1] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}} => [5,3,2,4,6,1] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => 4
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}} => [5,4,3,2,6,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}} => [4,3,2,6,5,1] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 3
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => {{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => {{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => {{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => {{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => {{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[] => [1] => {{1}} => [1] => 1

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 3\ q^{2} + q^{3}$

$F_{4} = q + 6\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = q + 10\ q^{2} + 21\ q^{3} + 9\ q^{4} + q^{5}$

Description

The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ .
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs

$(i, \pi(i))$ with

$i\leq\pi(i)$ .

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

Ringel

Description

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