Identifier
Values
0 => [2] => [1,1,0,0] => [1,2] => 0
1 => [1,1] => [1,0,1,0] => [2,1] => 0
00 => [3] => [1,1,1,0,0,0] => [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0] => [3,1,2] => 0
10 => [1,2] => [1,0,1,1,0,0] => [2,3,1] => 0
11 => [1,1,1] => [1,0,1,0,1,0] => [2,1,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 1
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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
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
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.