Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
St000056: Permutations ⟶ ℤ (values match St000234The number of global ascents of a permutation.)
Values
[1,0] => [1] => 1
[1,0,1,0] => [2,1] => 1
[1,1,0,0] => [1,2] => 2
[1,0,1,0,1,0] => [2,3,1] => 1
[1,0,1,1,0,0] => [2,1,3] => 2
[1,1,0,0,1,0] => [1,3,2] => 2
[1,1,0,1,0,0] => [3,1,2] => 1
[1,1,1,0,0,0] => [1,2,3] => 3
[1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,0] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,0] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,0] => [1,3,4,2] => 2
[1,1,0,0,1,1,0,0] => [1,3,2,4] => 3
[1,1,0,1,0,0,1,0] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,0] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => 3
[1,1,1,0,0,1,0,0] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 3
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => 1
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 3
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 3
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 3
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 4
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 3
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 1
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 4
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 2
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [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
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => 2
>>> Load all 196 entries. <<<
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Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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