Identifier
-
Mp00254:
Permutations
—Inverse fireworks map⟶
Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 0
[2,1] => [2,1] => [2,1] => ([],2) => 0
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 0
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3) => 0
[3,2,1] => [3,2,1] => [3,2,1] => ([],3) => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4) => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 0
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4) => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 0
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4) => 0
[2,4,1,3] => [2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[3,1,4,2] => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[3,2,4,1] => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[3,4,1,2] => [2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4) => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => 0
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 0
[1,4,2,5,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,4,3,5,2] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[1,4,5,2,3] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,4,5,3,2] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,5,3,2,4] => [1,5,3,2,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => 0
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[2,1,3,5,4] => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[2,1,4,3,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[2,1,4,5,3] => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[2,3,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[2,3,1,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[2,3,4,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
[2,3,4,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[2,3,5,1,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[2,3,5,4,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[2,4,1,3,5] => [2,4,1,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[2,4,1,5,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[2,4,3,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[2,4,3,5,1] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[2,4,5,1,3] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[2,4,5,3,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[2,5,3,1,4] => [1,5,3,2,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[3,1,4,2,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[3,1,4,5,2] => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[3,1,5,4,2] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[3,2,4,1,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[3,2,4,5,1] => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[3,2,5,1,4] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[3,2,5,4,1] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[3,4,1,5,2] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[3,4,2,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[3,4,2,5,1] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[3,4,5,1,2] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[3,4,5,2,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,5,2,1,4] => [3,5,2,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,5,4,1,2] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[4,1,2,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
[4,1,5,2,3] => [3,1,5,2,4] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[4,1,5,3,2] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
[4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
[4,2,5,1,3] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,2,5,3,1] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[4,3,1,2,5] => [4,3,1,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,3,5,1,2] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,3,5,2,1] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[4,5,1,2,3] => [3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[4,5,2,1,3] => [3,5,2,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
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Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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