Identifier
Values
[1,0,1,0] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0] => [2,3,1] => [3,1,2] => 0
[1,1,0,0,1,0] => [3,1,2] => [3,2,1] => 1
[1,1,0,1,0,0] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,1,4,3] => 1
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [4,3,1,2] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,1,2,4] => 0
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,2,1,3] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,1,4,2] => 1
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,3,2,1] => 2
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,3,2] => 2
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [5,3,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [5,4,3,1,2] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,2,1,3,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [3,1,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,2,1,3,4] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [3,1,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,2,4,1,3] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [4,2,1,5,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,3,1,4,2] => 2
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [5,3,2,1,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,5,3,2,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [4,3,2,1,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,4,3,2,1] => 3
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
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.