- FindStat

Identifier

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤ

Values

[1] => [1] => [1] => ([],1) => 1

[-1] => [1] => [1] => ([],1) => 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

[-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

>>> Load all 154 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

click to show known generating functions

$F_{1} = 2\ q$

$F_{2} = 8\ q$

$F_{3} = 48\ q$

Description

The number of connected components of the Hasse diagram for 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.

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.

Map

permutation

Description

The permutation obtained by forgetting the colours.