Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000054
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
Mp00252: Permutations restrictionPermutations
St000054: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => 1
[[.,.],.]
=> [1,2] => [2,1] => [1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1,3] => [2,1] => 2
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2] => 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => [2,1] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1,4] => [3,2,1] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1,2,4] => [3,1,2] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1,4] => [2,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => [2,1,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,1,3] => [2,1,3] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,2,3] => [1,2,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => [2,1,3] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,4,1,2] => [3,1,2] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,4,3] => [2,1,3] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,4,3] => [1,2,3] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [3,1,2] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => [2,3,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,1,2,5] => [4,3,1,2] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1] => 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,2,1,3,5] => [4,2,1,3] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3] => 4
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,4,2,1,5] => [3,4,2,1] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,4,1,2,5] => [3,4,1,2] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1] => 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1,5] => [2,3,4,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,2,4,5] => [3,1,2,4] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,1,4,5] => [2,3,1,4] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,3,2,1,4] => [3,2,1,4] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,3,1,2,4] => [3,1,2,4] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,2,3,1,4] => [2,3,1,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,2,1,3,4] => [2,1,3,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,2,3,4] => [1,2,3,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,5,2,1,4] => [3,2,1,4] => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,5,1,2,4] => [3,1,2,4] => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,5,2,1,3] => [4,2,1,3] => 4
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,1,2,3] => 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => [3,2,1,4] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => [2,3,1,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,5,1,3] => [4,2,1,3] => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,4,5,1,3] => [2,4,1,3] => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [3,4,5,1,2] => [3,4,1,2] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,4] => 3
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 [[St000703]] 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 $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$