- FindStat

Identifier

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000307: Posets ⟶ ℤ

Values

[[1,2]] => 0 => ([(0,1)],2) => 1
[[1],[2]] => 1 => ([(0,1)],2) => 1
[[1,2,3]] => 00 => ([(0,2),(2,1)],3) => 1
[[1,3],[2]] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[[1,2],[3]] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[[1],[2],[3]] => 11 => ([(0,2),(2,1)],3) => 1
[[1,2,3,4]] => 000 => ([(0,3),(2,1),(3,2)],4) => 1
[[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) => 3
[[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) => 3
[[1,2],[3,4]] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[[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) => 3
[[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) => 1
[[1,2,3,4,5]] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[1],[2],[3],[4],[5]] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[1,2,3,4,5,6]] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[1],[2],[3],[4],[5],[6]] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[[1,2,3,4,5,6,7]] => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[[1],[2],[3],[4],[5],[6],[7]] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1

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$F_{2} = 2\ q$

$F_{3} = 2\ q + 2\ q^{2}$

$F_{4} = 2\ q + 4\ q^{2} + 4\ q^{3}$

Description

The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset

$P$ . It sends an order ideal

$I$ to the order ideal generated by the minimal antichain of

$P \setminus I$ .

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

$v$ .