- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000019: Permutations ⟶ ℤ

Values

[1] => [1,0] => [2,1] => 1

[1,2] => [1,0,1,0] => [3,1,2] => 2

[2,1] => [1,1,0,0] => [2,3,1] => 2

[1,2,3] => [1,0,1,0,1,0] => [4,1,2,3] => 3

[1,3,2] => [1,0,1,1,0,0] => [3,1,4,2] => 3

[2,1,3] => [1,1,0,0,1,0] => [2,4,1,3] => 3

[2,3,1] => [1,1,0,1,0,0] => [4,3,1,2] => 3

[3,1,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3

[3,2,1] => [1,1,1,0,0,0] => [2,3,4,1] => 3

[1,2,3,4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4

[1,2,4,3] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 4

[1,3,2,4] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 4

[1,3,4,2] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 4

[1,4,2,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 4

[1,4,3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 4

[2,1,3,4] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 4

[2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 4

[2,3,1,4] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 4

[2,3,4,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 4

[2,4,1,3] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 4

[2,4,3,1] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 4

[3,1,2,4] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 4

[3,1,4,2] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4

[3,2,1,4] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 4

[3,2,4,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4

[3,4,1,2] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 4

[3,4,2,1] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 4

[4,1,2,3] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4

[4,1,3,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4

[4,2,1,3] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4

[4,2,3,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4

[4,3,1,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4

[4,3,2,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the support of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The connectivity set of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.

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

Ringel

Description

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