Identifier
Values
[1,0] => [1] => 0
[1,0,1,0] => [2,1] => 1
[1,1,0,0] => [1,2] => 0
[1,0,1,0,1,0] => [2,1,3] => 1
[1,0,1,1,0,0] => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => 1
[1,1,0,1,0,0] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0] => [2,4,1,3] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0] => [2,3,1,4] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => 2
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => 2
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => 2
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 3
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 1
>>> Load all 197 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [2,3,1,4,5,6] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [2,3,4,1,5,6] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5] => 3
[1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [3,4,1,6,2,5] => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => 2
[1,1,0,0,1,1,0,1,0,1,0,0] => [3,4,1,5,2,6] => 2
[1,1,0,0,1,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,5,6,2] => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [3,4,1,5,6,2] => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [3,4,5,1,6,2] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,5,2,6,4] => 3
[1,1,0,1,0,0,1,0,1,1,0,0] => [3,5,1,2,6,4] => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,5,6,2,4] => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [3,5,1,6,2,4] => 2
[1,1,0,1,0,0,1,1,1,0,0,0] => [3,5,6,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4,6] => 2
[1,1,0,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4,6] => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4,6] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4,6] => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [3,1,2,5,6,4] => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,2,5,6,4] => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,5,2,6,4] => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [1,3,5,6,2,4] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,6,2,4,5] => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [3,6,1,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,2,6,4,5] => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,2,6,4,5] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [1,3,6,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0,1,0] => [3,1,2,4,6,5] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,2,4,6,5] => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,4,2,6,5] => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [1,3,4,6,2,5] => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [3,1,2,4,5,6] => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [1,3,2,4,5,6] => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,4,2,5,6] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,4,5,2,6] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,3,4,5,6,2] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => 3
[1,1,1,0,0,0,1,0,1,1,0,0] => [4,5,1,2,6,3] => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [4,1,5,6,2,3] => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [4,5,1,6,2,3] => 2
[1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [4,1,5,2,3,6] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [4,5,1,2,3,6] => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [4,1,2,5,3,6] => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,3,6] => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [4,1,2,5,6,3] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,5,6,3] => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,5,2,6,3] => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,5,6,2,3] => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,6,2,3,5] => 2
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,6,1,2,3,5] => 1
[1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,2,6,3,5] => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => 2
[1,1,1,0,1,0,0,1,1,0,0,0] => [1,4,6,2,3,5] => 1
[1,1,1,0,1,0,1,0,0,0,1,0] => [4,1,2,3,6,5] => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => 2
[1,1,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5] => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,5,6] => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,5,6] => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [1,2,4,3,5,6] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,2,4,5,6,3] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [5,1,6,2,3,4] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,1,2,6,3,4] => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,6,3,4] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [1,5,6,2,3,4] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,1,2,3,6,4] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,3,6,4] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [1,2,5,3,6,4] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [1,2,5,6,3,4] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,4,6] => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [1,2,5,3,4,6] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,2,3,5,6,4] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,2,3,4,5] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [1,2,3,6,4,5] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[] => [] => 0
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
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.