{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "74JREVZVY3"
    },
    {
      "field": "instance_of",
      "target": "XDZQYX6AQ3"
    }
  ],
  "body": "\n## Summary\n\nAn Eulerian poset is a finite [[graded poset|1C7TKGTB2F]] with a least and a greatest element.\nIn it, every interval of positive length has as many elements of even rank as of odd rank.\nEquivalently, its [[Möbius function|2Y75S8NG5B]] is $\\mu(x, y) = (-1)^{\\rho(y) - \\rho(x)}$, where $\\rho$ is the rank.\nFace lattices of convex polytopes are Eulerian, and the condition encodes the Euler–Poincaré relation for their face numbers.\n\n## Explanations\n\n### L1 human\n\nCount the elements in an interval of the poset, sorted by their level.\nIn an Eulerian poset the levels with even numbers and the levels with odd numbers always hold the same total.\nThe subsets of a finite set form an Eulerian poset: a set with $k \\ge 1$ elements has as many subsets of even size as of odd size.\nThis is the same balance as $V - E + F = 2$ for a convex polyhedron.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A07",
      "52B05"
    ],
    "name": "Eulerian poset",
    "same_as": {
      "wikidata": "Q5409003",
      "wikipedia": "Eulerian poset"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Eulerian_poset"
      }
    ],
    "specializes": [
      "1C7TKGTB2F"
    ],
    "tag": "N1Q0BJAEFX"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/N1Q0BJAEFX/",
  "links": [
    {
      "field": "specializes",
      "target": "1C7TKGTB2F"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/N1Q0BJAEFX.md",
  "msc": [
    "06A07",
    "52B05"
  ],
  "name": "Eulerian poset",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N1Q/N1Q0BJAEFX.md",
  "status": null,
  "tag": "N1Q0BJAEFX",
  "uri": "m2://N1Q0BJAEFX",
  "url": "https://tamnd.github.io/math2/wiki/eulerian-poset/",
  "writing": {
    "tag": "N1Q0BJAEFX",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}