Balancing Extensions in Posets of Large Width
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916949272821760 |
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| author | Aires, Max Kahn, Jeff |
| author_facet | Aires, Max Kahn, Jeff |
| contents | We revisit classic balancing problems for linear extensions of a partially ordered set $P$, proving results that go far beyond many of the best earlier results on this topic. For example, with $p(x\prec y)$ the probability that $x$ precedes $y$ in a uniform linear extension, $δ_{xy} = \min\{p(x \prec y), p(y \prec x)\}$, and $δ(P)=\max δ_{xy}$, we show that $δ(P)$ tends to $1/2$ as $n := |P| \to\infty$ if $P$ has width $Ω(n)$ or $ω(\log(n))$ minimal elements, and is at least $1/e-o(1)$ if $P$ has width $ω(\sqrt{n})$ or height $o(n)$.
Motivated by both consequences for balance problems and intrinsic interest, we also consider several old and new parameters associated with $P$. Here, in addition to balance, we study relations between the parameters and suggest various questions that are thought to be worthy of further investigation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11549 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Balancing Extensions in Posets of Large Width Aires, Max Kahn, Jeff Combinatorics Probability 06A07, 60C05, 05A20 We revisit classic balancing problems for linear extensions of a partially ordered set $P$, proving results that go far beyond many of the best earlier results on this topic. For example, with $p(x\prec y)$ the probability that $x$ precedes $y$ in a uniform linear extension, $δ_{xy} = \min\{p(x \prec y), p(y \prec x)\}$, and $δ(P)=\max δ_{xy}$, we show that $δ(P)$ tends to $1/2$ as $n := |P| \to\infty$ if $P$ has width $Ω(n)$ or $ω(\log(n))$ minimal elements, and is at least $1/e-o(1)$ if $P$ has width $ω(\sqrt{n})$ or height $o(n)$. Motivated by both consequences for balance problems and intrinsic interest, we also consider several old and new parameters associated with $P$. Here, in addition to balance, we study relations between the parameters and suggest various questions that are thought to be worthy of further investigation. |
| title | Balancing Extensions in Posets of Large Width |
| topic | Combinatorics Probability 06A07, 60C05, 05A20 |
| url | https://arxiv.org/abs/2509.11549 |