- FindStat

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Identifier

St000717: Posets ⟶ ℤ

Values

([],2) => 1

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

([],3) => 1

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

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

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

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

([],4) => 1

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

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

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

([(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) => 1

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

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

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

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

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

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

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

([],5) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 1

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

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

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

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

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

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

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

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

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

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

([(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) => 2

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

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

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

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

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

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

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

([(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) => 1

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

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

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

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

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

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

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

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

([(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) => 1

([(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) => 1

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

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

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

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

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

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

([(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) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{4} = 7\ q + 5\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 31\ q + 18\ q^{2} + 9\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 184\ q + 80\ q^{2} + 34\ q^{3} + 14\ q^{4} + 5\ q^{5} + q^{6}$

Description

The number of ordinal summands of a poset.
The ordinal sum of two posets

$P$ and

$Q$ is the poset having elements

$(p,0)$ and

$(q,1)$ for

$p\in P$ and

$q\in Q$ , and relations

$(a,0) < (b,0)$ if

$a < b$ in

$P$ ,

$(a,1) < (b,1)$ if

$a < b$ in

$Q$ , and

$(a,0) < (b,1)$ .
This statistic is the length of the longest ordinal decomposition of a poset.

References

[1] wikipedia:Partially_ordered_set#Sums_of_partially_ordered_sets

Code

def statistic(P):
    return len(P.ordinal_summands())

Created

Mar 23, 2017 at 17:37 by Martin Rubey

Updated

Mar 23, 2017 at 21:02 by Martin Rubey