Identifier
-
Mp00255:
Decorated permutations
—lower permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤ
Values
[-,+] => [2,1] => [1,2] => ([(0,1)],2) => 1
[-,+,+] => [2,3,1] => [2,1,3] => ([(0,2),(1,2)],3) => 1
[-,-,+] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3) => 2
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 0
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 0
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 0
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 0
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 0
[-,+,4,3,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[-,-,4,3,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[-,3,2,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[-,3,2,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[-,3,4,2,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[-,4,2,3,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[-,4,+,2,+] => [3,2,5,1,4] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => 1
[-,4,-,2,+] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 0
[-,5,-,2,4] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
[-,5,-,+,2] => [4,2,1,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 0
[2,4,+,1,+] => [3,1,5,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[2,5,-,+,1] => [4,1,2,5,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[4,-,+,1,+] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => 1
[5,-,+,1,4] => [3,1,4,5,2] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[5,-,+,+,1] => [3,4,1,5,2] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
[5,+,-,+,1] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
[5,-,-,+,1] => [4,1,5,2,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[5,-,+,-,1] => [3,1,5,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[5,-,4,3,1] => [3,1,5,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
[5,3,2,+,1] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[-,-,+,+,+,+] => [3,4,5,6,1,2] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[-,+,-,+,+,+] => [2,4,5,6,1,3] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[-,+,+,-,+,+] => [2,3,5,6,1,4] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[-,+,+,+,-,+] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0
[-,-,-,+,+,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,-,+,-,+,+] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,-,+,+,-,+] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,-,-,+,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,-,+,-,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,+,-,-,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,-,-,-,+,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,-,-,+,-,+] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,-,+,-,-,+] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,+,-,-,-,+] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,-,-,-,-,+] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 0
[-,+,+,5,4,+] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0
[-,-,+,5,4,+] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,-,5,4,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,-,-,5,4,+] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,+,4,3,+,+] => [2,3,5,6,1,4] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[-,-,4,3,+,+] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,4,3,-,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,-,4,3,-,+] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,+,4,5,3,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,-,4,5,3,+] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,+,5,3,4,+] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0
[-,-,5,3,4,+] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,5,+,3,+] => [2,4,3,6,1,5] => [5,3,4,1,6,2] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6) => 1
[-,-,5,+,3,+] => [4,3,6,1,2,5] => [3,4,1,6,5,2] => ([(0,3),(1,2),(1,4),(1,5),(3,4),(3,5)],6) => 1
[-,+,5,-,3,+] => [2,3,6,1,5,4] => [5,4,1,6,2,3] => ([(0,5),(1,5),(2,3),(2,5),(3,4)],6) => 0
[-,-,5,-,3,+] => [3,6,1,2,5,4] => [4,1,6,5,2,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6) => 0
[-,+,6,-,3,5] => [2,3,5,1,6,4] => [5,4,2,6,1,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6) => 0
[-,-,6,-,3,5] => [3,5,1,2,6,4] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,+,6,-,+,3] => [2,5,3,1,6,4] => [5,2,4,6,1,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 0
[-,-,6,-,+,3] => [5,3,1,2,6,4] => [2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6) => 0
[-,3,2,+,+,+] => [2,4,5,6,1,3] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[-,3,2,-,+,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,3,2,+,-,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,3,2,-,-,+] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,3,2,5,4,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,3,4,2,+,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,3,4,2,-,+] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,3,4,5,2,+] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[-,3,5,2,4,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,3,5,+,2,+] => [4,2,6,1,3,5] => [3,5,1,6,4,2] => ([(0,3),(0,4),(1,2),(1,4),(1,5),(3,5)],6) => 0
[-,3,5,-,2,+] => [2,6,1,3,5,4] => [5,1,6,4,2,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[-,3,6,-,2,5] => [2,5,1,3,6,4] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6) => 0
[-,3,6,-,+,2] => [5,2,1,3,6,4] => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[-,4,2,3,+,+] => [2,3,5,6,1,4] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[-,4,2,3,-,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,4,2,5,3,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,4,+,2,+,+] => [3,2,5,6,1,4] => [4,5,2,1,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6) => 1
[-,4,-,2,+,+] => [2,5,6,1,4,3] => [5,2,1,6,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6) => 0
[-,4,+,2,-,+] => [3,2,6,1,4,5] => [4,5,1,6,3,2] => ([(0,4),(1,2),(1,3),(1,5),(4,5)],6) => 1
[-,4,-,2,-,+] => [2,6,1,4,3,5] => [5,1,6,3,4,2] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[-,4,+,5,2,+] => [3,2,6,1,4,5] => [4,5,1,6,3,2] => ([(0,4),(1,2),(1,3),(1,5),(4,5)],6) => 1
[-,4,-,5,2,+] => [2,6,1,4,3,5] => [5,1,6,3,4,2] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6) => 0
[-,4,5,2,3,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[-,4,5,3,2,+] => [3,2,6,1,4,5] => [4,5,1,6,3,2] => ([(0,4),(1,2),(1,3),(1,5),(4,5)],6) => 1
[-,5,2,3,4,+] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0
[-,5,2,+,3,+] => [2,4,3,6,1,5] => [5,3,4,1,6,2] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6) => 1
[-,5,2,-,3,+] => [2,3,6,1,5,4] => [5,4,1,6,2,3] => ([(0,5),(1,5),(2,3),(2,5),(3,4)],6) => 0
[-,5,+,2,4,+] => [3,2,4,6,1,5] => [4,5,3,1,6,2] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6) => 1
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Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
lower permutation
Description
The lower bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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