- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => 7

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

[1,1,1,1,1,2] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 3

[1,1,1,1,2,1] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 3

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

[1,1,1,2,1,1] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 3

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

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

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

[1,1,2,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 3

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

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

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

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

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

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

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

[1,2,1,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,6] => [6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,3,4,5,6] => 7

[2,1,1,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 3

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

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

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

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

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

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Description

The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.