Intersection statistics for antichains in minuscule posets
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912868100734976 |
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| author | Propp, James |
| author_facet | Propp, James |
| contents | For a finite poset $P$, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of $|A\cap A'|$ over all ordered pairs of antichains. For general posets this statistic appears to have little structure, but for the classical minuscule posets with uniform combinatorial models it admits closed-form expressions. Though the proofs are elementary and combinatorial, the resulting formulas admit a natural interpretation in terms of weight diagrams of minuscule representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intersection statistics for antichains in minuscule posets Propp, James Combinatorics 06A07 For a finite poset $P$, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of $|A\cap A'|$ over all ordered pairs of antichains. For general posets this statistic appears to have little structure, but for the classical minuscule posets with uniform combinatorial models it admits closed-form expressions. Though the proofs are elementary and combinatorial, the resulting formulas admit a natural interpretation in terms of weight diagrams of minuscule representations. |
| title | Intersection statistics for antichains in minuscule posets |
| topic | Combinatorics 06A07 |
| url | https://arxiv.org/abs/2601.21127 |