Identifier
Values
[[1,2]] => 0 => ([(0,1)],2) => 0
[[1],[2]] => 1 => ([(0,1)],2) => 0
[[1,2,3]] => 00 => ([(0,2),(2,1)],3) => 0
[[1,3],[2]] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[1,2],[3]] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[1],[2],[3]] => 11 => ([(0,2),(2,1)],3) => 0
[[1,2,3,4]] => 000 => ([(0,3),(2,1),(3,2)],4) => 0
[[1,3,4],[2]] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[[1,2,4],[3]] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[[1,2,3],[4]] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[[1,3],[2,4]] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[[1,2],[3,4]] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[[1,4],[2],[3]] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[[1,3],[2],[4]] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[[1,2],[3],[4]] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[[1],[2],[3],[4]] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[[1,2,3,4,5]] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[[1],[2],[3],[4],[5]] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[[1,2,3,4,5,6]] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[[1],[2],[3],[4],[5],[6]] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
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Description
The balance constant multiplied with the number of linear extensions of a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$
Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced.
Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Map
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.
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 $v$.