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  "body": "\n## Example\n\nYoung's lattice $\\mathbb{Y}$ is the set of all [[integer partitions|0JV30XN73Z]] $\\lambda = (\\lambda_1 \\ge \\lambda_2 \\ge \\cdots)$, including the empty partition.\nIt is ordered by $\\lambda \\le \\mu$ when $\\lambda_i \\le \\mu_i$ for all $i$, that is, when the [[Young diagram|096AHMB3P4]] of $\\lambda$ fits inside that of $\\mu$.\n\nIt is a [[distributive lattice|0HM0HQCK6N]].\nThe meet and join are $(\\lambda \\wedge \\mu)_i = \\min(\\lambda_i, \\mu_i)$ and $(\\lambda \\vee \\mu)_i = \\max(\\lambda_i, \\mu_i)$, the intersection and union of the diagrams.\nBoth are again weakly decreasing sequences, and $\\min$ and $\\max$ distribute over each other.\n\nIt is a [[graded poset|1C7TKGTB2F]] with rank function $\\rho(\\lambda) = \\abs{\\lambda} = \\lambda_1 + \\lambda_2 + \\cdots$.\nIndeed, $\\mu$ covers $\\lambda$ exactly when the diagram of $\\mu$ is that of $\\lambda$ with one more box.\n\n## Remarks\n\nThe saturated chains from $\\varnothing$ to $\\lambda$ add one box at a time, so they correspond to the standard [[Young tableaux|Q6TH9PH2R7]] of shape $\\lambda$.\nFor $\\lambda = (2, 1)$ there are two of them: $\\varnothing, (1), (2), (2, 1)$ and $\\varnothing, (1), (1, 1), (2, 1)$.\nThe partitions of $n$ form the rank $n$ level, and their number is the partition number $p(n)$.\n\n## Explanations\n\n### L1 human\n\nDraw each partition as a staircase of boxes in the top-left corner of a grid.\nOne staircase is below another if it fits inside it.\nGoing up one step in the lattice means adding one box, so climbing from nothing to a shape is a way of building that shape box by box.\n",
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