Identifier
Values
[1] => 10 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[2] => 100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 6
[1,1] => 110 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 7
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of posets with the same zeta polynomial.
The zeta polynomial $Z$ is the polynomial such that $Z(m)$ is the number of weakly increasing sequences $x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset.
See section 3.12 of [1].
Since
$$ Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k, $$
where $c_k$ is the number of chains of length $k$, this statistic is the same as the number of posets with the same chain polynomial.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
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 $v$.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.