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| In [[combinatorics|combinatorial]] [[mathematics]], the '''Lubell–Yamamoto–Meshalkin inequality''', more commonly known as the '''LYM inequality''', is an inequality on the sizes of sets in a [[Sperner family]], proved by {{harvtxt|Bollobás|1965}}, {{harvtxt|Lubell|1966}}, {{harvtxt|Meshalkin|1963}}, and {{harvtxt|Yamamoto|1954}}. It is named for the initials of three of its discoverers.
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| This inequality belongs to the field of [[combinatorics]] of sets, and has many applications in combinatorics. In particular, it can be used to prove [[Sperner's theorem]]. Its name is also used for similar inequalities.
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| ==Statement of the theorem==
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| Let ''U'' be an ''n''-element set, let ''A'' be a family of subsets of ''U'' such that no set in ''A'' is a subset of another set in ''A'', and let ''a<sub>k</sub>'' denote the number of sets of size ''k'' in ''A''. Then
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| : <math>\sum_{k=0}^n\frac{a_k}{{n \choose k}} \le 1.</math> | |
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| ==Lubell's proof==
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| {{harvtxt|Lubell|1966}} proves the Lubell–Yamamoto–Meshalkin inequality by a [[double counting (proof technique)|double counting argument]] in which he counts the [[permutation]]s of ''U'' in two different ways. First, by counting all permutations of ''U'' directly, one finds that there are ''n''! of them. But secondly, one can generate a permutation of ''U'' by selecting a set ''S'' in ''A'' and concatenating a permutation of the elements of ''S'' with a permutation of the nonmembers. If |''S''| = ''k'', it will be associated in this way with ''k''!(''n'' − ''k'')! permutations.
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| Each permutation can only be associated with a single set in ''A'', for if two prefixes of a permutation both formed sets in ''A'' then one would be a subset of the other. Therefore, the number of permutations that can be generated by this procedure is
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| :<math>\sum_{S\in A}|S|!(n-|S|)!=\sum_{k=0}^n a_k k! (n-k)!.</math> | |
| Since this number is at most the total number of all permutations,
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| :<math>\sum_{k=0}^n a_k k! (n-k)!\le n!.</math>
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| Finally dividing the above inequality by ''n''! leads to the result.
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| == References ==
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| *{{citation
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| | first = B. | last = Bollobás | authorlink = Béla Bollobás
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| | title = On generalized graphs
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| | journal = Acta Mathematica Academiae Scientiarum Hungaricae
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| | volume = 16 | issue = 3–4 | pages = 447–452 | year = 1965
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| | doi = 10.1007/BF01904851 |mr=0183653 }}.
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| *{{citation
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| | last = Lubell | first = D.
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| | year = 1966
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| | title = A short proof of Sperner's lemma
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| | journal = Journal of Combinatorial Theory
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| | volume = 1 | issue = 2 | pages = 299
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| | doi = 10.1016/S0021-9800(66)80035-2 |mr=0194348 }}.
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| *{{citation
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| | last = Meshalkin | first = L. D.
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| | year = 1963
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| | title = Generalization of Sperner's theorem on the number of subsets of a finite set
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| | journal = Theory of Probability and its Applications
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| | volume = 8 | issue = 2 | pages = 203–204
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| | doi = 10.1137/1108023 |mr=0150049 }}.
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| *{{citation
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| | last = Yamamoto | first = Koichi
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| | year = 1954
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| | title = Logarithmic order of free distributive lattice
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| | journal = Journal of the Mathematical Society of Japan
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| | volume = 6 | pages = 343–353
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| |mr=0067086 }}.
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| {{DEFAULTSORT:Lubell-Yamamoto-Meshalkin inequality}}
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| [[Category:Combinatorics]]
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| [[Category:Inequalities]]
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| [[Category:Order theory]]
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| [[Category:Set families]]
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| [[Category:Articles containing proofs]]
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