Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,1,2] => 1
[1,1,0,0,1,0] => [3,1,2] => [3,2,1] => 2
[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] => 2
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,2,1,3] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,2,1,4] => 2
[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] => 3
[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] => 2
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [4,3,1,2,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [5,3,1,2,4] => 2
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [2,1,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [5,4,3,1,2] => 3
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,1,2,5,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,4,1,2,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,2,1,3,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [3,1,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [3,1,5,2,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,2,4,1,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [5,4,2,1,3] => 3
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [3,1,5,4,2] => 3
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [3,2,1,5,4] => 3
[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] => 2
[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] => 3
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,3,1,4,2] => 3
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [5,3,2,1,4] => 3
[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] => 3
[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] => 4
[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
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [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] => [4,3,1,2,6,5] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [2,1,6,5,3,4] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [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] => [4,3,1,2,5,6] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,1,5,3,4,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [5,3,1,2,4,6] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [4,1,2,5,3,6] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [2,1,6,3,4,5] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [2,1,6,4,3,5] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [5,3,1,2,6,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [2,1,5,4,3,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [5,4,3,1,2,6] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [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] => [3,1,2,5,4,6] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,4,1,2,3,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [3,1,2,6,4,5] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [2,1,6,5,4,3] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [2,1,3,6,5,4] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [3,1,2,6,5,4] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [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] => [3,1,2,4,6,5] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [4,1,2,3,6,5] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => 2
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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].
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.
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.
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.
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.
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