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Identifier

St000525: Posets ⟶ ℤ

Values

([],2) => 1

([(0,1)],2) => 1

([],3) => 1

([(1,2)],3) => 1

([(0,1),(0,2)],3) => 2

([(0,2),(2,1)],3) => 1

([(0,2),(1,2)],3) => 2

([],4) => 1

([(2,3)],4) => 1

([(1,2),(1,3)],4) => 3

([(0,1),(0,2),(0,3)],4) => 3

([(0,2),(0,3),(3,1)],4) => 2

([(0,1),(0,2),(1,3),(2,3)],4) => 3

([(1,2),(2,3)],4) => 1

([(0,3),(3,1),(3,2)],4) => 3

([(1,3),(2,3)],4) => 3

([(0,3),(1,3),(3,2)],4) => 3

([(0,3),(1,3),(2,3)],4) => 3

([(0,3),(1,2)],4) => 3

([(0,3),(1,2),(1,3)],4) => 3

([(0,2),(0,3),(1,2),(1,3)],4) => 1

([(0,3),(2,1),(3,2)],4) => 1

([(0,3),(1,2),(2,3)],4) => 2

([],5) => 1

([(3,4)],5) => 1

([(2,3),(2,4)],5) => 3

([(1,2),(1,3),(1,4)],5) => 5

([(0,1),(0,2),(0,3),(0,4)],5) => 7

([(0,2),(0,3),(0,4),(4,1)],5) => 5

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 8

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5

([(1,3),(1,4),(4,2)],5) => 3

([(0,3),(0,4),(4,1),(4,2)],5) => 8

([(1,2),(1,3),(2,4),(3,4)],5) => 3

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 4

([(0,3),(0,4),(3,2),(4,1)],5) => 8

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 5

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 3

([(2,3),(3,4)],5) => 1

([(1,4),(4,2),(4,3)],5) => 3

([(0,4),(4,1),(4,2),(4,3)],5) => 5

([(2,4),(3,4)],5) => 3

([(1,4),(2,4),(4,3)],5) => 3

([(0,4),(1,4),(4,2),(4,3)],5) => 3

([(1,4),(2,4),(3,4)],5) => 5

([(0,4),(1,4),(2,4),(4,3)],5) => 5

([(0,4),(1,4),(2,4),(3,4)],5) => 7

([(0,4),(1,4),(2,3)],5) => 5

([(0,4),(1,3),(2,3),(2,4)],5) => 7

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,4),(1,4),(2,3),(4,2)],5) => 4

([(0,4),(1,3),(2,3),(3,4)],5) => 8

([(0,4),(1,4),(2,3),(2,4)],5) => 7

([(0,4),(1,4),(2,3),(3,4)],5) => 5

([(1,4),(2,3)],5) => 3

([(1,4),(2,3),(2,4)],5) => 5

([(0,4),(1,2),(1,4),(2,3)],5) => 5

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 5

([(1,3),(1,4),(2,3),(2,4)],5) => 7

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 3

([(0,4),(1,2),(1,4),(4,3)],5) => 8

([(0,4),(1,2),(1,3)],5) => 5

([(0,4),(1,2),(1,3),(1,4)],5) => 7

([(0,2),(0,4),(3,1),(4,3)],5) => 2

([(0,4),(1,2),(1,3),(3,4)],5) => 5

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 3

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 8

([(0,3),(0,4),(1,2),(1,4)],5) => 7

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 1

([(0,3),(1,2),(1,4),(3,4)],5) => 5

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2

([(1,4),(3,2),(4,3)],5) => 1

([(0,3),(3,4),(4,1),(4,2)],5) => 4

([(1,4),(2,3),(3,4)],5) => 3

([(0,4),(1,2),(2,4),(4,3)],5) => 3

([(0,3),(1,4),(4,2)],5) => 3

([(0,4),(3,2),(4,1),(4,3)],5) => 3

([(0,4),(1,2),(2,3),(2,4)],5) => 8

([(0,4),(2,3),(3,1),(4,2)],5) => 1

([(0,3),(1,2),(2,4),(3,4)],5) => 8

([(0,4),(1,2),(2,3),(3,4)],5) => 2

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 4

([],6) => 1

([(4,5)],6) => 1

([(3,4),(3,5)],6) => 3

([(2,3),(2,4),(2,5)],6) => 6

([(1,2),(1,3),(1,4),(1,5)],6) => 13

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 13

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 13

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 24

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => 18

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 18

([(1,3),(1,4),(1,5),(5,2)],6) => 9

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 24

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => 12

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 5

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 7

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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.

References

[1] Stanley, R. P. Enumerative combinatorics. Volume 1 MathSciNet:2868112

Code

@cached_function
def zeta_polynomial_multiplicities(n):
    l = [P.zeta_polynomial() for P in Posets(n)]
    return {p: l.count(p) for p in l}
    

def statistic(P):
    return zeta_polynomial_multiplicities(P.cardinality())[P.zeta_polynomial()]

Created

Jun 06, 2016 at 08:44 by Martin Rubey

Updated

Nov 08, 2017 at 19:01 by Martin Rubey