Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001890: Posets ⟶ ℤ
Values
{{1,2}} => [2,1] => [2,1] => ([],2) => 1
{{1},{2}} => [1,2] => [1,2] => ([(0,1)],2) => 1
{{1,2,3}} => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3) => 1
{{1,2},{3}} => [2,1,3] => [3,1,2] => ([(1,2)],3) => 1
{{1,3},{2}} => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 1
{{1},{2,3}} => [1,3,2] => [2,3,1] => ([(1,2)],3) => 1
{{1},{2},{3}} => [1,2,3] => [3,2,1] => ([],3) => 1
{{1,2,3,4}} => [2,3,4,1] => [4,2,1,3] => ([(1,3),(2,3)],4) => 1
{{1,2,3},{4}} => [2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => 1
{{1,2,4},{3}} => [2,4,3,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
{{1,2},{3,4}} => [2,1,4,3] => [4,1,3,2] => ([(1,2),(1,3)],4) => 1
{{1,2},{3},{4}} => [2,1,3,4] => [4,1,2,3] => ([(1,2),(2,3)],4) => 1
{{1,3,4},{2}} => [3,2,4,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 1
{{1,3},{2,4}} => [3,4,1,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 1
{{1,3},{2},{4}} => [3,2,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
{{1,4},{2,3}} => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
{{1},{2,3,4}} => [1,3,4,2] => [2,3,4,1] => ([(1,2),(2,3)],4) => 1
{{1},{2,3},{4}} => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 1
{{1,4},{2},{3}} => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 1
{{1},{2,4},{3}} => [1,4,3,2] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 1
{{1},{2},{3,4}} => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4) => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4) => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5) => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => 1
{{1,2,4,5},{3}} => [2,4,3,5,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5) => 1
{{1,2,4},{3,5}} => [2,4,5,1,3] => [5,3,4,2,1] => ([(3,4)],5) => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5) => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 1
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5) => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5) => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 1
{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5) => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5) => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 1
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1
{{1,4},{2,3,5}} => [4,3,5,1,2] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 1
{{1,5},{2,3,4}} => [5,3,4,2,1] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
{{1},{2,3,4,5}} => [1,3,4,5,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 1
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 1
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5) => 1
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 1
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5) => 1
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [5,4,1,3,2] => ([(2,3),(2,4)],5) => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5) => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 1
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The maximum magnitude of the Möbius function of a poset.
The Möbius function of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
The Möbius function of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Map
toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!