Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
St000470: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000325The width of the tree associated to a permutation.)
Values
[1,0] => [1] => 1
[1,0,1,0] => [2,1] => 2
[1,1,0,0] => [1,2] => 1
[1,0,1,0,1,0] => [2,1,3] => 2
[1,0,1,1,0,0] => [2,3,1] => 2
[1,1,0,0,1,0] => [3,1,2] => 2
[1,1,0,1,0,0] => [1,3,2] => 2
[1,1,1,0,0,0] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [2,1,4,3] => 3
[1,0,1,0,1,1,0,0] => [2,4,1,3] => 2
[1,0,1,1,0,0,1,0] => [2,1,3,4] => 2
[1,0,1,1,0,1,0,0] => [2,3,1,4] => 2
[1,0,1,1,1,0,0,0] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0] => [3,1,4,2] => 3
[1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0] => [1,3,2,4] => 2
[1,1,0,1,1,0,0,0] => [1,3,4,2] => 2
[1,1,1,0,0,0,1,0] => [4,1,2,3] => 2
[1,1,1,0,0,1,0,0] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0] => [1,2,4,3] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 3
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => 3
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => 3
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => 3
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => 3
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => 3
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => 2
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => 3
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => 3
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => 3
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => 3
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => 3
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => 3
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => 2
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => 3
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => 3
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => 2
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 2
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 4
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 2
>>> Load all 306 entries. <<<
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Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
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.
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.
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