{
  "id": "1509.06942",
  "version": "v1",
  "published": "2015-09-23T12:34:17.000Z",
  "updated": "2015-09-23T12:34:17.000Z",
  "title": "Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices",
  "authors": [
    "Henri Mühle"
  ],
  "comment": "27 pages, 5 pictures. Comments are welcome. In particular, I welcome any references to a combinatorial proof for the identity $\\sum_{j=0}^{k}\\binom{2j}{j}\\binom{2(k-j)}{k-j-1}=4^k-\\binom{2k+1}{k}$",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that the noncrossing partition lattices associated with the complex reflection groups $G(d,d,n)$ for $d,n\\geq 2$ admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the $k$ largest antichains does not exceed the sum of the $k$ largest ranks for all $k\\leq n$. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type $A$ and $B$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-23T12:34:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "20F55"
    ],
    "keywords": [
      "noncrossing partition lattice",
      "strong sperner property",
      "symmetric chain decompositions",
      "well-generated complex reflection group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150906942M"
    }
  }
}