# Improving the $$\frac{1}{3} - \frac{2}{3}$$ Conjecture for Width Two Posets

## Abstract

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset P of width 2. This constant is defined as δ(P) = max(x,y)∈P2 min{ℙ(xy), ℙ(yx)}, where ℙ(x≺y) is the probability x is less than y in a uniformly random linear extension of P. In particular, we show that if P is a width 2 poset that cannot be formed from the singleton poset and the three element poset with one relation using the operation of direct sum, then

$$\delta \left( P \right) \ge \frac{{ - 3 + 5\sqrt {17} }}{{52}} \approx 0.33876....$$

This partially answers a question of Brightwell (1999); a full resolution would require a proof of the $$\frac{1}{3} - \frac{2}{3}$$ Conjecture that if P is not totally ordered, then $$\delta \left( P \right) \ge \frac{1}{3}$$.

Furthermore, we construct a sequence of posets Tn of width 2 with δ(Tn) → β ≈ 0.348843…, giving an improvement over a construction of Chen (2017) and over the finite posets found by Peczarski (2017). Numerical work on small posets by Peczarski suggests the constant β may be optimal.

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## References

1. [1]

M. Aigner: A note on merging, Order 2 (1985), 257–264.

2. [2]

G. R. Brightwell, S. Felsner and W. T. Trotter: Balancing pairs and the cross product conjecture, Order 12 (1995), 327–349.

3. [3]

G. Brightwell: Balanced pairs in partial orders, Discrete Mathematics 201 (1999), 25–52.

4. [4]

G. R. Brightwell: Semiorders and the 1/3–2/3 conjecture, Order 5 (1989), 369–380.

5. [5]

J. Cardinal, S. Fiorini, G. Joret, R. M. Jungers and J. I. Munro: Sorting under partial information (without the ellipsoid algorithm), Combinatorica 33 (2013), 655–697.

6. [6]

E. Chen: A family of partially ordered sets with small balance constant, Electron. J. Combin. 25 Paper 4.43, 13, 2018.

7. [7]

S. Fiorini and S. Rexhep: Poset entropy versus number of linear extensions: the width-2 case, Order 33 (2016), 1–21.

8. [8]

P. C. Fishburn: On linear extension majority graphs of partial orders, Journal of Combinatorial Theory, Series B 21 (1976), 65–70.

9. [9]

M. L. Fredman: How good is the information theory bound in sorting? Theoretical Computer Science 1 (1976), 355–361.

10. [10]

S. G. Hoggar: Chromatic polynomials and logarithmic concavity, Journal of Combinatorial Theory, Series B 16 (1974), 248–254.

11. [11]

J. Kahn and J. H. Kim: Entropy and sorting, volume 51, 390–399. 1995. 24th Annual ACM Symposium on the Theory of Computing (Victoria, BC, 1992).

12. [12]

J. Kahn and M. Saks: Balancing poset extensions, Order 1 (1984), 113–126.

13. [13]

S. S. Kislitsyn: Finite partially ordered sets and their corresponding permutation sets, Math. Notes 4 (1968), 798–801.

14. [14]

N. Linial: The information-theoretic bound is good for merging, SIAM Journal on Computing 13 (1984), 795–801.

15. [15]

E. J. Olson and B. E. Sagan: On the 1/3–2/3 conjecture, Order 35 (2018), 581–596.

16. [16]

M. Peczarski: The Gold Partition Conjecture for 6-thin Posets, Order 25 (2008), 91–103.

17. [17]

M. Peczarski: The Worst Balanced Partially Ordered Sets—Ladders with Broken Rungs, Experimental Mathematics 28 (2019), 181–184.

18. [18]

W. T. Trotter, W. G. Gehrlein and P. C. Fishburn: Balance theorems for height-2 posets, Order 9 (1992), 43–53.

19. [19]

I. Zaguia: The 1/3–2/3 conjecture for N-free ordered sets, Electron. J. Combin. 19 Paper 29, 5, 2012.

20. [20]

I. Zaguia: The 1/3–2/3 conjecture for ordered sets whose cover graph is a forest, Order 36 (2019), 335–347.

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Correspondence to Ashwin Sah.

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Sah, A. Improving the $$\frac{1}{3} - \frac{2}{3}$$ Conjecture for Width Two Posets. Combinatorica 41, 99–126 (2021). https://doi.org/10.1007/s00493-020-4091-3