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  "body": "\n## Statement\n\nLet $t$ be a positive integer. Let $\\mathcal{P}(t)$ be the poset of [[partitions|0404R9P9J3]] of $t$ under the dominance order. A partition is written as the list of its parts.\n\nLet $V=(v_1,\\dots,v_m)$ and $W=(w_1,\\dots,w_n)$ be partitions of $t$, and suppose $V$ [[covers|EHXHC4ZNPA]] $W$ in $\\mathcal{P}(t)$.\n\nThen $n\\le m+1$, and there are integers $i\u003cj$ with $1\\le i\u003cj\\le n$ such that all of the following hold.\n\n1. $v_\\alpha=w_\\alpha$ for every $\\alpha\u003ci$.\n2. $v_\\beta=w_\\beta$ for every $\\beta$ with $j\u003c\\beta\\le m$.\n3. $v_i=w_i+1$.\n4. One of two cases occurs. Either $j\\le m$ and $w_j=v_j+1$, or $j=n=m+1$ and $w_j=1$.\n5. If $j\u003ei+1$, then $v_\\gamma=w_\\gamma=v_i-1$ for every $\\gamma$ with $i\u003c\\gamma\u003cj$.\n\n## Remarks\n\nIn words, a cover moves one unit from an earlier part to a later part. The parts strictly between the two changed positions all equal $v_i-1$. In case (b) of item 4, the unit goes into a new last part of size $1$, so $W$ has one more part than $V$.\n",
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        "locator": "source/s_kitchensink_zero-one-matrices.ptx:28",
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  "name": "Covering relation in the poset of partitions of an integer",
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