432:
192:
522:
51:. It is named for the initials of three of its discoverers. To include the initials of all four discoverers, it is sometimes referred to as the
299:
119:
206:
589:
443:
293:
then one would be a subset of the other. Therefore, the number of permutations that can be generated by this procedure is
546:
708:
693:
622:
Meshalkin, L. D. (1963), "Generalization of
Sperner's theorem on the number of subsets of a finite set",
703:
698:
63:
541:
573:
662:
631:
598:
555:
229:! of them. But secondly, one can generate a permutation (i.e., an order) of the elements of
62:
of sets, and has many applications in combinatorics. In particular, it can be used to prove
676:
643:
612:
569:
672:
639:
608:
565:
32:
603:
687:
577:
59:
17:
210:
20:
667:
653:
Yamamoto, Koichi (1954), "Logarithmic order of free distributive lattice",
427:{\displaystyle \sum _{S\in A}|S|!(n-|S|)!=\sum _{k=0}^{n}a_{k}k!(n-k)!.}
560:
635:
437:
Since this number is at most the total number of all permutations,
187:{\displaystyle \sum _{k=0}^{n}{\frac {a_{k}}{n \choose k}}\leq 1.}
285:. Each permutation may only be associated with a single set in
217:
in two different ways. First, by counting all permutations of
273:)! permutations, and in each of them the image of the first
289:, for if two prefixes of a permutation both formed sets in
587:
Lubell, D. (1966), "A short proof of
Sperner's lemma",
446:
302:
205:
proves the Lubell–Yamamoto–Meshalkin inequality by a
122:
517:{\displaystyle \sum _{k=0}^{n}a_{k}k!(n-k)!\leq n!.}
516:
426:
186:
66:. Its name is also used for similar inequalities.
547:Acta Mathematica Academiae Scientiarum Hungaricae
171:
158:
31:, is an inequality on the sizes of sets in a
8:
655:Journal of the Mathematical Society of Japan
624:Theory of Probability and Its Applications
666:
602:
559:
527:Finally dividing the above inequality by
472:
462:
451:
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391:
381:
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352:
344:
327:
319:
307:
301:
170:
157:
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144:
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127:
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44:
58:This inequality belongs to the field of
48:
36:
241:and choosing a map that sends {1, … , |
202:
40:
225:} directly, one finds that there are
7:
25:Lubell–Yamamoto–Meshalkin inequality
162:
105:denote the number of sets of size
14:
544:(1965), "On generalized graphs",
590:Journal of Combinatorial Theory
261:is associated in this way with
496:
484:
415:
403:
357:
353:
345:
335:
328:
320:
94:is a subset of another set in
1:
604:10.1016/S0021-9800(66)80035-2
27:, more commonly known as the
725:
86:be a family of subsets of
207:double counting argument
70:Statement of the theorem
531:! leads to the result.
221:identified with {1, …,
209:in which he counts the
518:
467:
428:
386:
188:
143:
668:10.2969/jmsj/00630343
519:
447:
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366:
197:
189:
123:
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300:
120:
90:such that no set in
269: −
233:by selecting a set
561:10.1007/BF01904851
514:
424:
318:
184:
82:-element set, let
303:
176:
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64:Sperner's theorem
716:
709:Families of sets
679:
670:
661:(3–4): 343–353,
646:
615:
606:
580:
563:
554:(3–4): 447–452,
523:
521:
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45:Meshalkin (1963)
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636:10.1137/1108023
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537:
468:
442:
441:
387:
298:
297:
200:
156:
146:
118:
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103:
72:
53:YBLM inequality
49:Yamamoto (1954)
37:Bollobás (1965)
12:
11:
5:
722:
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712:
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706:
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686:
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649:
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630:(2): 203–204,
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365:
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359:
355:
351:
347:
343:
340:
337:
334:
330:
326:
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316:
313:
310:
306:
253:| =
199:
198:Lubell's proof
196:
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183:
180:
173:
168:
165:
160:
153:
149:
141:
136:
133:
130:
126:
101:
71:
68:
33:Sperner family
29:LYM inequality
13:
10:
9:
6:
4:
3:
2:
721:
710:
707:
705:
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694:Combinatorics
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349:
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296:
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268:
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204:
203:Lubell (1966)
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116:
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60:combinatorics
56:
54:
50:
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42:
41:Lubell (1966)
38:
34:
30:
26:
22:
19:
18:combinatorial
704:Order theory
699:Inequalities
658:
654:
627:
623:
594:
588:
551:
545:
542:Bollobás, B.
528:
526:
436:
290:
286:
282:
278:
277:elements of
274:
270:
266:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
211:permutations
201:
110:
106:
99:
95:
91:
87:
83:
79:
75:
73:
57:
52:
35:, proved by
28:
24:
15:
281:is exactly
21:mathematics
688:Categories
597:(2): 299,
535:References
257:, the set
98:, and let
578:122892253
503:≤
491:−
449:∑
410:−
368:∑
342:−
312:∈
305:∑
179:≤
125:∑
677:0067086
644:0150049
613:0194348
570:0183653
245:| } to
113:. Then
47:, and
675:
642:
611:
576:
568:
249:. If |
78:be an
23:, the
574:S2CID
74:Let
663:doi
632:doi
599:doi
556:doi
237:in
213:of
109:in
16:In
690::
673:MR
671:,
657:,
640:MR
638:,
626:,
609:MR
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593:,
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566:MR
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265:!(
182:1.
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680:.
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634::
628:8
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601::
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581:.
558::
529:n
512:.
509:!
506:n
500:!
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494:k
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485:(
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479:k
474:k
470:a
464:n
459:0
456:=
453:k
422:.
419:!
416:)
413:k
407:n
404:(
401:!
398:k
393:k
389:a
383:n
378:0
375:=
372:k
364:=
361:!
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354:|
350:S
346:|
339:n
336:(
333:!
329:|
325:S
321:|
315:A
309:S
291:A
287:A
283:S
279:U
275:k
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219:U
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159:(
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148:a
140:n
135:0
132:=
129:k
111:A
107:k
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100:a
96:A
92:A
88:U
84:A
80:n
76:U
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