Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed permutations ⟶ ℤ
Values
[[1]] => [1] => [1] => [1] => 0
[[1,2]] => [1,2] => [2,1] => [2,1] => 0
[[1],[2]] => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[[1,3],[2]] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[[1,2],[3]] => [3,1,2] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2
[[1,3,4],[2]] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[[1,2,4],[3]] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 2
[[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 2
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2
[[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
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Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, St001866The nesting alignments of a signed permutation., an alignment of type EN, St001867The number of alignments of type EN of a signed permutation., or an alignment of type NE, St001868The number of alignments of type NE of a signed permutation..
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, St001862The number of crossings of a signed permutation., let \operatorname{wex} be the number of weak excedances, St001863The number of weak excedances of a signed permutation., and let \operatorname{neg} be the number of negative entries, St001429The number of negative entries in a signed permutation.. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, St001866The nesting alignments of a signed permutation., an alignment of type EN, St001867The number of alignments of type EN of a signed permutation., or an alignment of type NE, St001868The number of alignments of type NE of a signed permutation..
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, St001862The number of crossings of a signed permutation., let \operatorname{wex} be the number of weak excedances, St001863The number of weak excedances of a signed permutation., and let \operatorname{neg} be the number of negative entries, St001429The number of negative entries in a signed permutation.. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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