Identifier
Values
[1] => [1,0] => [1] => 0
[1,1] => [1,0,1,0] => [2,1] => 0
[2] => [1,1,0,0] => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0] => [2,1,3] => 0
[1,2] => [1,0,1,1,0,0] => [2,3,1] => 0
[2,1] => [1,1,0,0,1,0] => [3,1,2] => 0
[3] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5] => 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,5,6,2] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,5,1,2,6,3] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [4,1,5,6,2,3] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,1,6,2,3,4] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
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Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.