Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
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] => [2,3,1] => [3,1,2] => 1
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [3,1,2] => [3,2,1] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,3,1,2] => 2
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,2,1,3] => 2
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [3,1,4,2] => 2
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,2,1,4] => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => 1
[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] => [4,1,2,3] => [4,3,2,1] => 3
[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] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,1,2,5,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [5,4,1,2,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [5,3,1,2,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [4,1,2,5,3] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [4,3,1,2,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,5,4,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [5,4,3,1,2] => 3
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,5,4,2,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,2,1,3,4] => 2
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,2,1,3,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [3,1,5,2,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,2,4,1,3] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [3,1,4,2,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,2,1,5,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [5,4,2,1,3] => 3
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [3,1,5,4,2] => 3
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,3,2,4] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [5,3,2,1,4] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [4,2,1,5,3] => 3
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [5,3,1,4,2] => 3
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,3,2,1,5] => 3
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[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] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,1,2,3,6,5] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,1,2,6,4,5] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [3,1,2,5,4,6] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,4,1,2,3,6] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,1,2,4,6,5] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [3,1,2,6,5,4] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,3,4,5] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,1,5,3,4,6] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [2,1,6,5,3,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [5,3,1,2,4,6] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [4,1,2,6,3,5] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [4,1,2,5,3,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [4,3,1,2,6,5] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [4,3,1,2,5,6] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,6,4,3,5] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [2,1,5,4,3,6] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [5,3,1,2,6,4] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [5,4,3,1,2,6] => 3
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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
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.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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