Identifier
-
Mp00163:
Signed permutations
—permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤ
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[1,-2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[-1,2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[-1,-2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[2,-1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[-2,1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[-2,-1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[-1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[-1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[-1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[-1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,-3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,-3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-1,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-1,3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-1,-3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-1,-3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,-1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,-1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,-1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,-1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,-3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,-3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,-3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-2,-3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,-1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,-1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[-3,1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[-3,1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[-3,-1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[-3,-1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,-2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,-2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-3,2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-3,2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-3,-2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[-3,-2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,-2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,-2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[-1,-2,-3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,-1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,-1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,-1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[4,-1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,-1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,-1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,-1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[-4,-1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,-3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,-2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
>>> Load all 152 entries. <<<
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
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.
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 permutation obtained by forgetting the colours.
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!