Identifier
Values
[1,2] => 0 => 0 => ([(0,1)],2) => 1
[2,1] => 1 => 1 => ([(0,1)],2) => 1
[1,2,3] => 00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,2] => 01 => 00 => ([(0,2),(2,1)],3) => 1
[2,1,3] => 10 => 11 => ([(0,2),(2,1)],3) => 1
[2,3,1] => 01 => 00 => ([(0,2),(2,1)],3) => 1
[3,1,2] => 10 => 11 => ([(0,2),(2,1)],3) => 1
[3,2,1] => 11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,2,4] => 010 => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[1,4,2,3] => 010 => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[2,1,4,3] => 101 => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[2,3,1,4] => 010 => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[2,4,1,3] => 010 => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[3,1,4,2] => 101 => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[3,2,4,1] => 101 => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[3,4,1,2] => 010 => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,3,2] => 101 => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,3,1] => 101 => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[1,3,2,5,4] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,4,2,5,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,4,3,5,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,2,4,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,3,4,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,4,3,5] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,5,3,4] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,1,5,4] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,1,5,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,4,3,5,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,1,4,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,5,3,4,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,4,2,5] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,5,2,4] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,4,1,5] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,5,1,4] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,1,5,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,4,2,5,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,1,4,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,5,2,4,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,3,2,5] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,5,2,3] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,3,1,5] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,5,1,3] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,5,1,2] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,1,3,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,5,2,3,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,3,2,4] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,4,2,3] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,3,1,4] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,4,1,3] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,4,1,2] => 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],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 number of connected components of the Hasse diagram for the poset.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.