Identifier
-
Mp00253:
Decorated permutations
—permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000080: Posets ⟶ ℤ (values match St000528The height of a poset.)
Values
[+] => [1] => [1] => ([],1) => 0
[-] => [1] => [1] => ([],1) => 0
[+,+] => [1,2] => [1,2] => ([(0,1)],2) => 1
[-,+] => [1,2] => [1,2] => ([(0,1)],2) => 1
[+,-] => [1,2] => [1,2] => ([(0,1)],2) => 1
[-,-] => [1,2] => [1,2] => ([(0,1)],2) => 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[+,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[-,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 2
[3,+,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,-,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[+,4,2,3] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[-,4,2,3] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[2,3,4,1] => [2,3,4,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,4,1,3] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,4,+,1] => [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[2,4,-,1] => [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[3,1,2,+] => [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[3,1,2,-] => [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 3
[4,1,+,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,1,-,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,+,+,1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,-,+,1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,+,-,1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,-,-,1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[-,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[5,1,2,3,4] => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[+,+,+,+,+,+] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[-,+,+,+,+,+] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[+,-,+,+,+,+] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[+,+,-,+,+,+] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[+,+,+,-,+,+] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
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Description
The rank of the poset.
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
permutation
Description
The underlying permutation of the decorated permutation.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
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