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Identifier

St001105: Posets ⟶ ℤ

Values

([],2) => 2

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

([],3) => 6

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

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

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

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

([],4) => 24

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],5) => 120

([(3,4)],5) => 24

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 10

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 12

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

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

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

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

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

([(1,4),(2,3),(2,4)],5) => 11

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

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

([(1,3),(1,4),(2,3),(2,4)],5) => 16

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

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

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

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

([(0,4),(1,2),(1,3),(1,4)],5) => 10

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

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

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

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

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

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 12

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

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

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

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

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

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

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

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

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

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

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

([(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) => 2

([],6) => 720

([(4,5)],6) => 120

([(3,4),(3,5)],6) => 120

([(2,3),(2,4),(2,5)],6) => 120

([(1,2),(1,3),(1,4),(1,5)],6) => 120

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 120

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

([(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) => 24

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

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

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

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

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

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

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Description

The number of greedy linear extensions of a poset.
A linear extension of a poset $P$ with elements $\{x_1,\dots,x_n\}$ is greedy, if it can be obtained by the following algorithm:

Step 1. Choose a minimal element $x_1$.
Step 2. Suppose $X=\{x_1,\dots,x_i\}$ have been chosen. If there is at least one minimal element of $P\setminus X$ which is greater than $x_i$ then choose $x_{i+1}$ to be any such minimal element; otherwise, choose $x_{i+1}$ to be any minimal element of $P\setminus X$.

This statistic records the number of greedy linear extensions.

References

[1] Brualdi, R. A., Jung, H. C., Trotter, Jr., W. T. On the poset of all posets on $n$ elements MathSciNet:1275415

Code

def statistic(P):
    return sum(1 for i in P._hasse_diagram.greedy_linear_extensions_iterator())

Created

Feb 15, 2018 at 21:19 by Martin Rubey

Updated

Feb 15, 2018 at 21:19 by Martin Rubey