4096:
3494:
4091:{\displaystyle {\begin{aligned}\left(\!\!{4 \choose 18}\!\!\right)&={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3},\\&={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},\\&={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;1\cdot 2\cdot 3\quad } }},\\&={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.\end{aligned}}}
2508:
6165:
5594:
3343: • • • • • • | • • | • • • | • • • • • • •
5381:
6059:
operates on multisets and returns identical records. For instance, consider "SELECT name from
Student". In the case that there are multiple records with name "Sara" in the student table, all of them are shown. That means the result of an SQL query is a multiset; if the result were instead a set, the
6075:
used multisets as a device to investigate the properties of families of sets. He wrote, "The notion of a set takes no account of multiple occurrence of any one of its members, and yet it is just this kind of information that is frequently of importance. We need only think of the set of roots of a
6004:
4644:
4503:
2989:
5024:
5764:
5589:{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .}
5156:
360:, or units." These and similar collections of objects can be regarded as multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged.
5811:
4508:
4338:
4322:
1760:
2991:
where the second expression is as a binomial coefficient; many authors in fact avoid separate notation and just write binomial coefficients. So, the number of such multisets is the same as the number of subsets of cardinality
3094:
2765:
1111:
1290:, those normally can be excluded from the context. On the other hand, the latter notation is coherent with the fact that the prime factorization of a positive integer is a uniquely defined multiset, as asserted by the
4211:
3499:
2471:
1636:
6055:. For instance, multisets are often used to implement relations in database systems. In particular, a table (without a primary key) works as a multiset, because it can have multiple identical records. Similarly,
2209:
2075:
4879:
2326:
3489:
2597:
1197:
2677:
3166:
1941:
5649:
1426:. In this article the multiplicities are considered to be finite, so that no element occurs infinitely many times in the family; even in an infinite multiset, the multiplicities are finite numbers.
5802:, where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for
4872:
4797:
3246:
the equality of multiset coefficients and binomial coefficients given above involves representing multisets in the following way. First, consider the notation for multisets that would represent
5344:
5213:
761:
5654:
5040:
378:
Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician
990:
363:
Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names. For instance, they were important in early
564:
877:
4216:
2741:
4696:
1773:
The usual operations of sets may be extended to multisets by using the multiplicity function, in a similar way to using the indicator function for subsets. In the following,
1647:
1481:
1251:
181:
These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to
3013:
1457:
that does not belong to a given multiset is said to have a multiplicity 0 in this multiset. This extends the multiplicity function of the multiset to a function from
1424:
1284:
807:
1840:
1810:
7069:
5289:
4111:
1555:
1531:
687:, its multiplicity as a root of the minimal polynomial is the size of the largest Jordan block, and its geometric multiplicity is the number of Jordan blocks.
2491:), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an
2365:
1560:
2109:
1975:
5999:{\displaystyle (1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )},}
4639:{\displaystyle \left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.}
4498:{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0}
1498:, and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the
2235:
1014:
3385:
219:
denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset
3099:
5598:
With this definition one has a generalization of the negative binomial formula (with one of the variables set to 1), which justifies calling the
581:, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be
1872:
375:. A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set).
7297:
7292:
7271:
7135:
6806:
4813:
6678:
Syropoulos, Apostolos (2000). "Mathematics of multisets". In Calude, Cristian; Paun, Gheorghe; Rozenberg, Grzegorz; Salomaa, Arto (eds.).
4741:
2984:{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},}
632:
408:
Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example,
483:
in the multiset. Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.
6491:
2488:
6861:
6836:
6469:
6428:
1291:
1116:
2627:
2521:
4100:
From the relation between binomial coefficients and multiset coefficients, it follows that the number of multisets of cardinality
6457:
3370:
vertical lines among the 18 + 4 − 1 characters, and is thus the number of subsets of cardinality 4 − 1 of a set of cardinality
3366:. The number of vertical lines is 4 − 1. The number of multisets of cardinality 18 is then the number of ways to arrange the
5601:
6930:
5019:{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).}
2748:
2603:
590:
6380:
513:
31:
5296:
5165:
1429:
It is possible to extend the definition of a multiset by allowing multiplicities of individual elements to be infinite
6395:
By a set (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Gansen) M of definite and
2507:
67:
of that element in the multiset. As a consequence, an infinite number of multisets exist that contain only elements
2484:. This is equivalent to saying that their intersection is the empty multiset or that their sum equals their union.
725:
62:
6389:(in German). xlvi, xlix. New York Dover Publications (1954 English translation): 481–512, 207–246. Archived from
6060:
repetitive records in the result set would have been eliminated. Another application of multisets is in modeling
1484:
628:
285:
3007:. The analogy with binomial coefficients can be stressed by writing the numerator in the above expression as a
1299:
937:
640:
631:. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the
348:
Wayne
Blizard traced multisets back to the very origin of numbers, arguing that "in ancient times, the number
6928:
Libkin, L.; Wong, L. (1995). "On representation and querying incomplete information in databases with bags".
6570:
Singh, D.; Ibrahim, A. M.; Yohanna, T.; Singh, J. N. (2007). "An overview of the applications of multisets".
809:, especially when considering submultisets. This article is restricted to finite, positive multiplicities.)
7089:
6085:
2752:
2337:
644:
417:
2751:
in which the multiset coefficients occur. Multiset coefficients should not be confused with the unrelated
242:
or "size" of a multiset is the sum of the multiplicities of all its elements. For example, in the multiset
6959:
6142:
5791:
5216:
3008:
636:
602:
452:
425:
823:
6703:
6385:
6169:
6065:
2744:
1287:
58:
2702:
6100:
Different generalizations of multisets have been introduced, studied and applied to solving problems.
5366:
The multiplicative formula allows the definition of multiset coefficients to be extended by replacing
4651:
6325:
2680:
2496:
2492:
665:). These three multiplicities define three multisets of eigenvalues, which may be all different: Let
624:
620:
440:
300:
by many centuries. Knuth himself attributes the first study of multisets to the Indian mathematician
6356:
seen as a generalized binomial coefficient equals the extreme right-hand side of the above equation.
5759:{\displaystyle (1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.}
5151:{\displaystyle \sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.}
2495:
number of the given multisets, while in the second sum we consider all possible intersections of an
6048:
5795:
5034:
4332:
2756:
1382:
817:
598:
508:
6486:
6420:
1464:
7242:
7063:
7009:
6822:
6720:
6604:
6158:
3243:
2333:
1499:
1217:
680:
578:
574:
387:
6530:
7267:
7131:
6857:
6832:
6802:
6465:
6424:
54:
1396:
1256:
792:
30:
This article is about the mathematical concept. For the computer science data structure, see
7183:
7123:
7098:
7036:
6999:
6968:
6939:
6890:
6747:
6712:
6683:
6680:
Multiset
Processing, Mathematical, Computer Science, and Molecular Computing Points of View
6657:
6500:
6412:
6148:
3362:. The number of characters including both dots and vertical lines used in this notation is
1815:
402:
372:
1788:
6089:
6032:
6006:
and formulas such as these can be used to prove identities for the multiset coefficients.
5231:
4317:{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{k+1 \choose n-1}\!\!\right).}
1430:
409:
383:
6413:
187:, the order in which elements are listed does not matter in discriminating multisets, so
5236:
1755:{\displaystyle |A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)}
1433:
instead of positive integers, but not all properties carry over to this generalization.
1332:
A multiset corresponds to an ordinary set if the multiplicity of every element is 1. An
7204:
7056:
Proceedings of the 2nd
International Symposium on Methodologies for Intelligent Systems
6795:
6589:
Angelelli, I. (1965). "Leibniz's misunderstanding of
Nizolius' notion of 'multitudo'".
6461:
5803:
5351:
5227:
2329:
1540:
1516:
1450:
1333:
594:
472:
379:
301:
6164:
7286:
7103:
7084:
6972:
6943:
6910:
6278:
6121:
6044:
6027:
are zero, and the infinite series becomes a finite sum. However, for other values of
5769:
3378:
characters, which is the number of subsets of cardinality 18 of a set of cardinality
3170:
For example, there are 4 multisets of cardinality 3 with elements taken from the set
2481:
7013:
931:}. This notation is however not commonly used; more compact notations are employed.
6449:
6376:
6072:
1286:
If the elements of the multiset are numbers, a confusion is possible with ordinary
700:
504:
492:
391:
293:
282:
are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6.
6608:
4806:
does not appear, then our original multiset is equal to a multiset of cardinality
4698:
be the source set. There is always exactly one (empty) multiset of size 0, and if
3096:
to match the expression of binomial coefficients using a falling factorial power:
6682:. Lecture Notes in Computer Science. Vol. 2235. Springer. pp. 347–358.
4723:
with elements from might or might not contain any instance of the final element
6639:
6390:
6105:
5375:
3089:{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},}
395:
308:
around 1150. Other names have been proposed or used for this concept, including
305:
239:
38:
7004:
6987:
6505:
1106:{\displaystyle \left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad }
589:. In the latter case it has a solution of multiplicity 2. More generally, the
7054:
Grzymala-Busse, J. (1987). "Learning from examples based on rough multisets".
7040:
6701:
Whitney, Hassler (1933). "Characteristic
Functions and the Algebra of Logic".
6383:[contributions to the founding of the theory of transfinite numbers].
6153:
6061:
993:
616:
357:
7127:
6751:
6687:
6068:. As such, the entity that specifies the edges is a multiset, and not a set.
4738:
of elements from , and every such multiset can arise, which gives a total of
2511:
1767:
1362:
390:
found the number of multiset permutations when one element can be repeated.
7151:
Alkhazaleh, S.; Salleh, A. R.; Hassan, N. (2011). "Soft
Multisets Theory".
6895:
6878:
3374:. Equivalently, it is the number of ways to arrange the 18 dots among the
2487:
There is an inclusion–exclusion principle for finite multisets (similar to
7188:
7171:
4206:{\displaystyle \left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.}
386:(1498–1576) contains another early reference to the concept of multisets.
5159:
1295:
432:
17:
2336:
structure on the finite multisets in a given universe. This monoid is a
7122:. Lecture Notes in Computer Science. Vol. 2235. pp. 225–235.
6724:
2683:; it is used for instance in (Stanley, 1997), and could be pronounced "
2466:{\displaystyle m_{C}(x)=\max(m_{A}(x)-m_{B}(x),0)\quad \forall x\in U.}
1631:{\displaystyle \operatorname {Supp} (A):=\{x\in U\mid m_{A}(x)>0\}.}
1537:
is the underlying set of the multiset. Using the multiplicity function
1202:
where upper indices equal to 1 are omitted. For example, the multiset {
421:
6331:
6288:
6234:
6231:) if viewed as an ordinary binomial coefficient since it evaluates to
6192:
6104:
Real-valued multisets (in which multiplicity of an element can be any
6043:
Multisets have various applications. They are becoming fundamental in
1487:
between these functions and the multisets that have their elements in
7247:
6826:
3491:
thus is the value of the multiset coefficient and its equivalencies:
2204:{\displaystyle m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.}
2070:{\displaystyle m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.}
1503:
499:. Here the underlying set of elements is the set of prime factors of
296:. However, the concept of multisets predates the coinage of the word
7222:
Burgin, Mark (1992). "On the concept of a multiset in cybernetics".
6716:
6552:
Rulifson, J. F.; Derkson, J. A.; Waldinger, R. J. (November 1972).
6064:. In multigraphs there can be multiple edges between any two given
4705:
there are no larger multisets, which gives the initial conditions.
424:
value: positive, negative or zero). Monro (1987) investigated the
7118:
Miyamoto, S. (2001). "Fuzzy
Multisets and Their Generalizations".
6957:
Blizard, Wayne D. (1989). "Real-valued
Multisets and Fuzzy Sets".
6913:; Wong, L. (1994). "Some properties of query languages for bags".
1494:
This extended multiplicity function is commonly called simply the
1381:. In this view the underlying set of the multiset is given by the
183:
57:
that, unlike a set, allows for multiple instances for each of its
6740:
Zeitschrift für
Mathematische Logik und Grundlagen der Mathematik
623:, whose multiplicity is usually defined as their multiplicity as
491:
One of the simplest and most natural examples is the multiset of
382:
circa 1150, who described permutations of multisets. The work of
2321:{\displaystyle m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.}
1644:
if its support is finite, or, equivalently, if its cardinality
61:. The number of instances given for each element is called the
7205:"Theory of Named Sets as a Foundational Basis for Mathematics"
6056:
3484:{\displaystyle {4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,}
2762:
The value of multiset coefficients can be given explicitly as
2592:{\textstyle {7 \choose 3}=\left(\!\!{5 \choose 3}\!\!\right).}
1192:{\displaystyle \quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},}
6915:
Proceedings of the Workshop on Database Programming Languages
5362:
Generalization and connection to the negative binomial series
2672:{\displaystyle \textstyle \left(\!\!{n \choose k}\!\!\right)}
3161:{\displaystyle {n \choose k}={n^{\underline {k}} \over k!}.}
394:
published a general rule for multiset permutations in 1675.
7266:. Mathematics Research Developments. Nova Science Pub Inc.
7241:
Burgin, Mark (2004). "Unified Foundations of Mathematics".
6047:. Multisets have become an important tool in the theory of
364:
5350:, it and the generating function are well defined for any
1936:{\displaystyle m_{A}(x)\leq m_{B}(x)\quad \forall x\in U.}
1441:
Elements of a multiset are generally taken in a fixed set
5644:{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)}
4648:
The above recurrence may be interpreted as follows. Let
722:
of the multiset, formed from its distinct elements, and
6132:
Named sets (unification of all generalizations of sets)
2612:, with elements taken from a finite set of cardinality
367:
languages, such as QA4, where they were referred to as
6381:"beiträge zur begründung der transfiniten Mengenlehre"
4574:
4474:
2707:
2631:
2524:
771:– that is, the number of occurrences – of the element
5814:
5657:
5604:
5384:
5299:
5239:
5168:
5043:
4882:
4867:{\displaystyle \left(\!\!{n-1 \choose k}\!\!\right).}
4816:
4744:
4654:
4511:
4341:
4219:
4114:
3497:
3388:
3102:
3016:
2768:
2705:
2630:
2368:
2238:
2112:
1978:
1875:
1818:
1791:
1650:
1563:
1543:
1519:
1467:
1399:
1259:
1220:
1119:
1017:
940:
826:
795:
728:
573:
A related example is the multiset of solutions of an
516:
4792:{\displaystyle \left(\!\!{n \choose k-1}\!\!\right)}
1770:
support (underlying set), and thus a cardinality 0.
75:, but vary in the multiplicities of their elements:
6277:does work in this case because the numerator is an
5226:indeterminates. Thus, the above series is also the
5158:As multisets are in one-to-one correspondence with
5037:of the multiset coefficients is very simple, being
1385:of the family, and the multiplicity of any element
6794:
6554:QA4: A Procedural Calculus for Intuitive Reasoning
5998:
5758:
5643:
5588:
5339:{\displaystyle \left(\!\!{n \choose d}\!\!\right)}
5338:
5283:
5208:{\displaystyle \left(\!\!{n \choose d}\!\!\right)}
5207:
5150:
5018:
4866:
4791:
4690:
4638:
4497:
4316:
4205:
4090:
3483:
3160:
3088:
2983:
2735:
2671:
2591:
2516:and 3-multisets with elements from a 5-set (right)
2465:
2320:
2203:
2069:
1935:
1834:
1804:
1754:
1630:
1549:
1525:
1475:
1418:
1278:
1245:
1191:
1105:
984:
871:
801:
755:
683:that has a single eigenvalue. Its multiplicity is
558:
6071:There are also other applications. For instance,
5737:
5736:
5730:
5717:
5713:
5712:
5635:
5634:
5628:
5615:
5611:
5610:
5415:
5414:
5408:
5395:
5391:
5390:
5330:
5329:
5323:
5310:
5306:
5305:
5199:
5198:
5192:
5179:
5175:
5174:
5095:
5094:
5088:
5075:
5071:
5070:
5007:
5006:
5000:
4979:
4975:
4974:
4960:
4959:
4953:
4932:
4928:
4927:
4913:
4912:
4906:
4893:
4889:
4888:
4855:
4854:
4848:
4827:
4823:
4822:
4783:
4782:
4776:
4755:
4751:
4750:
4731:once, one is left with a multiset of cardinality
4611:
4610:
4604:
4591:
4587:
4586:
4542:
4541:
4535:
4522:
4518:
4517:
4466:
4465:
4459:
4438:
4434:
4433:
4419:
4418:
4412:
4391:
4387:
4386:
4372:
4371:
4365:
4352:
4348:
4347:
4305:
4304:
4298:
4269:
4265:
4264:
4250:
4249:
4243:
4230:
4226:
4225:
4194:
4176:
4145:
4144:
4138:
4125:
4121:
4120:
3616:
3603:
3561:
3548:
3532:
3531:
3525:
3512:
3508:
3507:
3466:
3439:
3427:
3392:
3119:
3106:
3047:
3046:
3040:
3027:
3023:
3022:
2838:
2811:
2799:
2798:
2792:
2779:
2775:
2774:
2662:
2661:
2638:
2637:
2580:
2579:
2556:
2555:
6738:Monro, G. P. (1987). "The Concept of Multiset".
6120:Multisets whose multiplicity is any real-valued
2747:that involves binomial coefficients, there is a
2391:
2135:
2001:
789:(It is also possible to allow multiplicity 0 or
2679:, a notation that is meant to resemble that of
7058:. Charlotte, North Carolina. pp. 325–332.
7027:Yager, R. R. (1986). "On the Theory of Bags".
6673:
6671:
6524:
6522:
6520:
6518:
6516:
6379:; Jourdain, Philip E.B. (Translator) (1895).
2723:
2710:
2655:
2642:
2573:
2560:
2541:
2528:
756:{\displaystyle m\colon A\to \mathbb {Z} ^{+}}
401:Multisets appeared explicitly in the work of
8:
6419:. Jones & Bartlett Publishers. pp.
4685:
4667:
2624:. This number is written by some authors as
2328:It may be viewed as a generalization of the
1622:
1582:
1240:
1221:
979:
947:
866:
827:
767:to the set of positive integers, giving the
398:explained this rule in more detail in 1685.
6856:. Vol. 2. Cambridge University Press.
6831:. Vol. 1. Cambridge University Press.
6565:
6563:
6444:
6442:
6440:
1368:, may define a multiset, sometimes written
7085:"Sets with a negative numbers of elements"
7068:: CS1 maint: location missing publisher (
6556:(Technical report). SRI International. 73.
4335:for multiset coefficients may be given as
4009:
4004:
3939:
3934:
7246:
7187:
7102:
7003:
6894:
6504:
5972:
5944:
5931:
5906:
5884:
5856:
5831:
5813:
5772:formula is valid for all complex numbers
5747:
5729:
5716:
5714:
5701:
5690:
5674:
5656:
5627:
5614:
5612:
5603:
5575:
5571:
5570:
5559:
5454:
5430:
5424:
5407:
5394:
5392:
5383:
5322:
5309:
5307:
5298:
5269:
5250:
5238:
5191:
5178:
5176:
5167:
5136:
5114:
5105:
5087:
5074:
5072:
5059:
5048:
5042:
4999:
4978:
4976:
4952:
4931:
4929:
4905:
4892:
4890:
4881:
4847:
4826:
4824:
4815:
4775:
4754:
4752:
4743:
4653:
4603:
4590:
4588:
4573:
4565:
4564:
4534:
4521:
4519:
4510:
4473:
4458:
4437:
4435:
4411:
4390:
4388:
4364:
4351:
4349:
4340:
4297:
4268:
4266:
4242:
4229:
4227:
4218:
4193:
4175:
4173:
4167:
4137:
4124:
4122:
4113:
4043:
4005:
3958:
3935:
3886:
3778:
3777:
3775:
3755:
3736:
3641:
3640:
3638:
3635:
3615:
3602:
3600:
3587:
3570:
3560:
3547:
3545:
3524:
3511:
3509:
3498:
3496:
3465:
3438:
3436:
3426:
3391:
3389:
3387:
3355:made of elements of a set of cardinality
3134:
3128:
3118:
3105:
3103:
3101:
3062:
3056:
3039:
3026:
3024:
3015:
2907:
2882:
2847:
2837:
2810:
2808:
2791:
2778:
2776:
2767:
2722:
2709:
2706:
2704:
2654:
2641:
2639:
2629:
2572:
2559:
2557:
2540:
2527:
2525:
2523:
2423:
2401:
2373:
2367:
2287:
2265:
2243:
2237:
2167:
2145:
2117:
2111:
2033:
2011:
1983:
1977:
1902:
1880:
1874:
1823:
1817:
1796:
1790:
1737:
1721:
1699:
1671:
1659:
1651:
1649:
1601:
1562:
1542:
1518:
1483:of non-negative integers. This defines a
1469:
1468:
1466:
1404:
1398:
1264:
1258:
1228:
1219:
1175:
1164:
1159:
1141:
1130:
1125:
1118:
1083:
1072:
1067:
1043:
1032:
1027:
1016:
985:{\displaystyle A=\{a_{1},\ldots ,a_{n}\}}
973:
954:
939:
825:
794:
747:
743:
742:
727:
547:
537:
527:
515:
352:was often represented by a collection of
27:Mathematical set with repetitions allowed
7029:International Journal of General Systems
4810:with elements from , of which there are
2506:
53:) is a modification of the concept of a
6883:Journal of Computer and System Sciences
6368:
6182:
2608:The number of multisets of cardinality
7170:Alkhazaleh, S.; Salleh, A. R. (2012).
7061:
4727:. If it does appear, then by removing
1506:, and shares some properties with it.
608:always form a multiset of cardinality
6879:"Towards tractable algebras for bags"
7:
6764:Cf., for instance, Richard Brualdi,
6664:. Braunschweig: Vieweg. p. 114.
6531:"The Development of Multiset Theory"
615:A special case of the above are the
559:{\displaystyle 120=2^{3}3^{1}5^{1},}
7209:Structures in Mathematical Theories
6917:. Springer Verlag. pp. 97–114.
6782:. New York/Berlin: Springer Verlag.
6662:Was sind und was sollen die Zahlen?
5790:. It can also be interpreted as an
5215:is also the number of monomials of
3863:
3857:
3851:
3845:
3839:
3833:
3827:
3821:
3815:
3809:
3803:
3797:
3791:
3785:
3779:
3726:
3720:
3714:
3708:
3702:
3696:
3690:
3684:
3678:
3672:
3666:
3660:
3654:
3648:
3642:
2514:between 3-subsets of a 7-set (left)
872:{\displaystyle \{(a,m(a)):a\in A\}}
7211:. San Sebastian. pp. 417–420.
6992:Notre Dame Journal of Formal Logic
6591:Notre Dame Journal of Formal Logic
6492:Notre Dame Journal of Formal Logic
6031:, including positive integers and
5721:
5702:
5619:
5399:
5314:
5183:
5079:
5060:
4983:
4936:
4897:
4831:
4759:
4595:
4526:
4442:
4395:
4356:
4273:
4234:
4180:
4129:
3607:
3552:
3516:
3443:
3396:
3348:This is a multiset of cardinality
3110:
3031:
2815:
2783:
2714:
2646:
2564:
2532:
2448:
2303:
2186:
2052:
1918:
1781:are multisets in a given universe
879:) allows for writing the multiset
796:
270:the multiplicities of the members
99:, each having multiplicity 1 when
25:
3776:
3639:
2736:{\displaystyle {\tbinom {n}{k}}.}
1292:fundamental theorem of arithmetic
6326:generalized binomial coefficient
6163:
5651:negative binomial coefficients:
4708:Now, consider the case in which
4691:{\displaystyle :=\{1,\dots ,n\}}
4022:
4016:
4010:
3952:
3946:
3940:
3768:
3762:
3756:
3749:
3743:
3737:
7224:Cybernetics and System Analysis
6572:Novi Sad Journal of Mathematics
6458:The Art of Computer Programming
6051:, which often uses the synonym
5911:
5905:
5558:
4626:
4580:
4572:
4557:
4472:
4025:
2499:number of the given multisets.
2447:
2340:, with the universe as a basis.
2302:
2185:
2086:(called, in some contexts, the
2051:
1952:(called, in some contexts, the
1917:
1766:is the unique multiset with an
1437:Basic properties and operations
1120:
1102:
661:is an eigenvalue of the matrix
6931:Information Processing Letters
6877:Grumbach, S.; Milo, T (1996).
6460:. Vol. 2 (3rd ed.).
6168:Learning materials related to
5988:
5976:
5969:
5956:
5941:
5928:
5915:
5912:
5900:
5888:
5881:
5868:
5853:
5840:
5828:
5815:
5671:
5658:
5546:
5534:
5531:
5519:
5511:
5493:
5487:
5475:
5472:
5460:
5275:
5243:
5133:
5120:
4661:
4655:
4164:
4154:
2964:
2946:
2940:
2928:
2925:
2913:
2895:
2883:
2868:
2850:
2749:negative binomial distribution
2604:Stars and bars (combinatorics)
2444:
2435:
2429:
2413:
2407:
2394:
2385:
2379:
2299:
2293:
2277:
2271:
2255:
2249:
2182:
2179:
2173:
2157:
2151:
2138:
2129:
2123:
2048:
2045:
2039:
2023:
2017:
2004:
1995:
1989:
1914:
1908:
1892:
1886:
1785:, with multiplicity functions
1749:
1743:
1711:
1705:
1690:
1684:
1660:
1652:
1613:
1607:
1576:
1570:
1389:is the number of index values
1181:
1168:
1147:
1134:
1089:
1076:
1049:
1036:
851:
848:
842:
830:
775:in the multiset as the number
738:
699:may be formally defined as an
591:fundamental theorem of algebra
443:between elements "of the same
1:
7298:Factorial and binomial topics
7176:Abstract and Applied Analysis
7153:Applied Mathematical Sciences
1309:corresponds to the multiset {
32:Multiset (abstract data type)
7293:Basic concepts in set theory
7172:"Fuzzy Soft Multiset Theory"
7104:10.1016/0001-8708(92)90011-9
6973:10.1016/0165-0114(89)90218-2
6944:10.1016/0020-0190(95)00154-5
6852:Stanley, Richard P. (1999).
6797:Combinatorics of Finite Sets
5435:
4719:. A multiset of cardinality
3208:of cardinality 3 in the set
3067:
1476:{\displaystyle \mathbb {N} }
1449:, which is often the set of
1302:; for example, the monomial
6801:. Oxford: Clarendon Press.
2362:with multiplicity function
2232:with multiplicity function
2106:with multiplicity function
1972:with multiplicity function
1246:{\displaystyle \{a^{2},b\}}
292:in the 1970s, according to
7314:
6986:Blizard, Wayne D. (1990).
6766:Introductory Combinatorics
6529:Blizard, Wayne D. (1991).
6035:, the series is infinite.
2616:, is sometimes called the
2601:
1557:, it is characterized as
820:(the set of ordered pairs
812:Representing the function
639:, which is defined as the
503:. For example, the number
29:
7041:10.1080/03081078608934952
6854:Enumerative Combinatorics
6828:Enumerative Combinatorics
6626:Elemens des Mathematiques
6485:Blizard, Wayne D (1989).
6013:is a nonpositive integer
5577: and arbitrary
2518:So this illustrates that
1485:one-to-one correspondence
1008:is often represented as
629:characteristic polynomial
566:which gives the multiset
471:) from a multiset to the
306:permutations of multisets
286:Nicolaas Govert de Bruijn
177:both have multiplicity 3.
7128:10.1007/3-540-45523-X_11
7005:10.1305/ndjfl/1093635499
6752:10.1002/malq.19870330212
6688:10.1007/3-540-45523-X_17
6646:. London: John Playford.
6506:10.1305/ndjfl/1093634995
6454:Seminumerical Algorithms
6224:(where necessarily also
4104:in a set of cardinality
2996:of a set of cardinality
2753:multinomial coefficients
1113:sometimes simplified to
459:. He also introduced a
418:characteristic functions
134:has multiplicity 2, and
7090:Advances in Mathematics
6628:. Paris: André Pralard.
6411:Hein, James L. (2003).
6170:Partitions of multisets
5370:by an arbitrary number
2338:free commutative monoid
2092:greatest common divisor
1419:{\displaystyle a_{i}=x}
1279:{\displaystyle a^{2}b.}
802:{\displaystyle \infty }
451:between multisets as a
431:of multisets and their
91:contains only elements
6960:Fuzzy Sets and Systems
6896:10.1006/jcss.1996.0042
6624:Prestet, Jean (1675).
6250:, however the formula
6145:as multiplicity analog
6143:Frequency (statistics)
6017:, then all terms with
6000:
5760:
5706:
5645:
5590:
5340:
5285:
5209:
5152:
5064:
5020:
4868:
4793:
4692:
4640:
4499:
4318:
4207:
4092:
3485:
3162:
3090:
3009:rising factorial power
2985:
2737:
2673:
2599:
2593:
2480:if their supports are
2467:
2332:of sets. It defines a
2322:
2205:
2071:
1958:lowest common multiple
1937:
1836:
1835:{\displaystyle m_{B}.}
1806:
1756:
1632:
1551:
1527:
1477:
1420:
1280:
1247:
1193:
1107:
986:
873:
803:
757:
637:geometric multiplicity
560:
111:is seen as a multiset.
7262:Burgin, Mark (2011).
7203:Burgin, Mark (1990).
6988:"Negative Membership"
6793:Anderson, I. (1987).
6704:Annals of Mathematics
6644:A treatise of algebra
6386:Mathematische Annalen
6001:
5761:
5686:
5646:
5591:
5341:
5286:
5210:
5153:
5044:
5021:
4869:
4794:
4693:
4641:
4500:
4319:
4208:
4093:
3486:
3163:
3091:
2986:
2745:binomial distribution
2738:
2681:binomial coefficients
2674:
2594:
2510:
2468:
2323:
2206:
2072:
1938:
1837:
1807:
1805:{\displaystyle m_{A}}
1757:
1633:
1552:
1528:
1496:multiplicity function
1478:
1445:, sometimes called a
1421:
1288:arithmetic operations
1281:
1248:
1194:
1108:
987:
874:
804:
758:
561:
371:a term attributed to
7264:Theory of Named Sets
6780:Combinatorial Theory
6613:. Rome: Corbelletti.
6610:Musurgia Universalis
6415:Discrete mathematics
6324:if interpreted as a
6313:does make sense for
6129:Soft fuzzy multisets
6049:relational databases
5812:
5655:
5602:
5382:
5297:
5237:
5166:
5041:
4880:
4814:
4742:
4652:
4509:
4339:
4217:
4112:
3495:
3386:
3100:
3014:
2766:
2703:
2628:
2618:multiset coefficient
2522:
2366:
2236:
2110:
1976:
1873:
1816:
1789:
1648:
1561:
1541:
1517:
1465:
1397:
1257:
1218:
1117:
1015:
938:
907:}, and the multiset
824:
793:
726:
514:
495:of a natural number
463: : a function
441:equivalence relation
235:can be denoted by .
7189:10.1155/2012/350603
7120:Multiset Processing
6823:Stanley, Richard P.
6778:Aigner, M. (1979).
6605:Kircher, Athanasius
5796:formal power series
5346:is a polynomial in
5035:generating function
4333:recurrence relation
4327:Recurrence relation
3204:. There are also 4
2757:multinomial theorem
1185:
1151:
1093:
1053:
763:is a function from
599:polynomial equation
509:prime factorization
138:has multiplicity 1.
6217:does not work for
6159:Bag-of-words model
5996:
5756:
5641:
5586:
5336:
5284:{\displaystyle k.}
5281:
5205:
5148:
5016:
4864:
4789:
4688:
4636:
4578:
4495:
4478:
4314:
4203:
4088:
4086:
3868:
3731:
3481:
3242:One simple way to
3212:of cardinality 4 (
3174:of cardinality 2 (
3158:
3142:
3086:
2981:
2755:that occur in the
2733:
2728:
2669:
2668:
2600:
2589:
2503:Counting multisets
2476:Two multisets are
2463:
2334:commutative monoid
2318:
2201:
2067:
1933:
1832:
1802:
1752:
1732:
1694:
1628:
1547:
1523:
1500:indicator function
1473:
1416:
1276:
1243:
1189:
1155:
1121:
1103:
1063:
1023:
982:
869:
799:
753:
681:Jordan normal form
633:minimal polynomial
579:quadratic equation
575:algebraic equation
556:
388:Athanasius Kircher
7273:978-1-61122-788-8
7137:978-3-540-43063-6
7083:Loeb, D. (1992).
6808:978-0-19-853367-2
6658:Dedekind, Richard
6487:"Multiset theory"
6399:objects m (p.85)
6353:
6352:
6310:
6309:
6247:
6246:
6214:
6213:
5909:
5728:
5626:
5578:
5562:
5556:
5449:
5438:
5406:
5321:
5190:
5143:
5086:
5029:Generating series
4998:
4951:
4904:
4846:
4774:
4602:
4577:
4533:
4477:
4457:
4410:
4363:
4296:
4241:
4192:
4136:
4079:
4028:
3871:
3614:
3595:
3559:
3523:
3464:
3425:
3338:s) in this form:
3153:
3135:
3117:
3081:
3070:
3038:
2976:
2902:
2836:
2790:
2721:
2653:
2571:
2539:
1717:
1667:
1550:{\displaystyle m}
1526:{\displaystyle A}
1361:varies over some
1298:is a multiset of
1214:} may be written
593:asserts that the
585:, or it could be
439:as a set with an
412:(1933) described
16:(Redirected from
7305:
7278:
7277:
7259:
7253:
7252:
7250:
7238:
7232:
7231:
7219:
7213:
7212:
7200:
7194:
7193:
7191:
7167:
7161:
7160:
7159:(72): 3561–3573.
7148:
7142:
7141:
7115:
7109:
7108:
7106:
7080:
7074:
7073:
7067:
7059:
7051:
7045:
7044:
7024:
7018:
7017:
7007:
6983:
6977:
6976:
6954:
6948:
6947:
6925:
6919:
6918:
6907:
6901:
6900:
6898:
6874:
6868:
6867:
6849:
6843:
6842:
6819:
6813:
6812:
6800:
6790:
6784:
6783:
6775:
6769:
6762:
6756:
6755:
6735:
6729:
6728:
6698:
6692:
6691:
6675:
6666:
6665:
6654:
6648:
6647:
6636:
6630:
6629:
6621:
6615:
6614:
6601:
6595:
6594:
6586:
6580:
6579:
6567:
6558:
6557:
6549:
6543:
6542:
6526:
6511:
6510:
6508:
6482:
6476:
6475:
6450:Knuth, Donald E.
6446:
6435:
6434:
6418:
6408:
6402:
6401:
6373:
6357:
6355:
6332:
6323:
6312:
6289:
6284:
6276:
6249:
6235:
6230:
6223:
6216:
6193:
6187:
6167:
6033:rational numbers
6030:
6026:
6016:
6012:
6005:
6003:
6002:
5997:
5992:
5991:
5952:
5951:
5939:
5938:
5910:
5907:
5904:
5903:
5864:
5863:
5839:
5838:
5789:
5787:
5765:
5763:
5762:
5757:
5752:
5751:
5742:
5738:
5735:
5734:
5733:
5720:
5705:
5700:
5682:
5681:
5650:
5648:
5647:
5642:
5640:
5636:
5633:
5632:
5631:
5618:
5595:
5593:
5592:
5587:
5579:
5576:
5574:
5563:
5560:
5557:
5555:
5514:
5455:
5450:
5448:
5440:
5439:
5431:
5425:
5420:
5416:
5413:
5412:
5411:
5398:
5373:
5369:
5357:
5349:
5345:
5343:
5342:
5337:
5335:
5331:
5328:
5327:
5326:
5313:
5290:
5288:
5287:
5282:
5274:
5273:
5255:
5254:
5225:
5221:
5214:
5212:
5211:
5206:
5204:
5200:
5197:
5196:
5195:
5182:
5157:
5155:
5154:
5149:
5144:
5142:
5141:
5140:
5115:
5110:
5109:
5100:
5096:
5093:
5092:
5091:
5078:
5063:
5058:
5025:
5023:
5022:
5017:
5012:
5008:
5005:
5004:
5003:
4994:
4982:
4965:
4961:
4958:
4957:
4956:
4950:
4935:
4918:
4914:
4911:
4910:
4909:
4896:
4873:
4871:
4870:
4865:
4860:
4856:
4853:
4852:
4851:
4842:
4830:
4809:
4805:
4798:
4796:
4795:
4790:
4788:
4784:
4781:
4780:
4779:
4773:
4758:
4737:
4730:
4726:
4722:
4718:
4704:
4697:
4695:
4694:
4689:
4645:
4643:
4642:
4637:
4616:
4612:
4609:
4608:
4607:
4594:
4579:
4575:
4568:
4547:
4543:
4540:
4539:
4538:
4525:
4504:
4502:
4501:
4496:
4479:
4475:
4471:
4467:
4464:
4463:
4462:
4453:
4441:
4424:
4420:
4417:
4416:
4415:
4409:
4394:
4377:
4373:
4370:
4369:
4368:
4355:
4323:
4321:
4320:
4315:
4310:
4306:
4303:
4302:
4301:
4295:
4284:
4272:
4255:
4251:
4248:
4247:
4246:
4233:
4212:
4210:
4209:
4204:
4199:
4198:
4197:
4188:
4179:
4172:
4171:
4150:
4146:
4143:
4142:
4141:
4128:
4107:
4103:
4097:
4095:
4094:
4089:
4087:
4080:
4078:
4061:
4044:
4036:
4029:
4027:
4026:
3956:
3955:
3887:
3879:
3872:
3870:
3869:
3867:
3866:
3771:
3753:
3752:
3732:
3730:
3729:
3636:
3628:
3621:
3620:
3619:
3606:
3596:
3594:
3579:
3571:
3566:
3565:
3564:
3551:
3537:
3533:
3530:
3529:
3528:
3515:
3490:
3488:
3487:
3482:
3471:
3470:
3469:
3460:
3442:
3432:
3431:
3430:
3424:
3413:
3395:
3381:
3377:
3373:
3369:
3365:
3361:
3354:
3344:
3337:
3333:
3329:
3325:
3321:
3238:
3234:
3230:
3226:
3222:
3211:
3203:
3199:
3195:
3191:
3187:
3180:
3173:
3167:
3165:
3164:
3159:
3154:
3152:
3144:
3143:
3129:
3124:
3123:
3122:
3109:
3095:
3093:
3092:
3087:
3082:
3080:
3072:
3071:
3063:
3057:
3052:
3048:
3045:
3044:
3043:
3030:
3006:
2995:
2990:
2988:
2987:
2982:
2977:
2975:
2967:
2908:
2903:
2901:
2874:
2848:
2843:
2842:
2841:
2832:
2814:
2804:
2800:
2797:
2796:
2795:
2782:
2742:
2740:
2739:
2734:
2729:
2727:
2726:
2713:
2698:
2694:
2690:
2686:
2678:
2676:
2675:
2670:
2667:
2663:
2660:
2659:
2658:
2645:
2615:
2611:
2598:
2596:
2595:
2590:
2585:
2581:
2578:
2577:
2576:
2563:
2546:
2545:
2544:
2531:
2489:the one for sets
2472:
2470:
2469:
2464:
2428:
2427:
2406:
2405:
2378:
2377:
2361:
2358:is the multiset
2357:
2353:
2327:
2325:
2324:
2319:
2292:
2291:
2270:
2269:
2248:
2247:
2231:
2228:is the multiset
2227:
2223:
2210:
2208:
2207:
2202:
2172:
2171:
2150:
2149:
2122:
2121:
2105:
2102:is the multiset
2101:
2097:
2076:
2074:
2073:
2068:
2038:
2037:
2016:
2015:
1988:
1987:
1971:
1968:is the multiset
1967:
1963:
1942:
1940:
1939:
1934:
1907:
1906:
1885:
1884:
1868:
1858:
1850:
1841:
1839:
1838:
1833:
1828:
1827:
1811:
1809:
1808:
1803:
1801:
1800:
1784:
1780:
1776:
1761:
1759:
1758:
1753:
1742:
1741:
1731:
1704:
1703:
1693:
1663:
1655:
1637:
1635:
1634:
1629:
1606:
1605:
1556:
1554:
1553:
1548:
1536:
1532:
1530:
1529:
1524:
1490:
1482:
1480:
1479:
1474:
1472:
1460:
1456:
1453:. An element of
1444:
1425:
1423:
1422:
1417:
1409:
1408:
1392:
1388:
1380:
1360:
1356:
1285:
1283:
1282:
1277:
1269:
1268:
1252:
1250:
1249:
1244:
1233:
1232:
1198:
1196:
1195:
1190:
1184:
1180:
1179:
1163:
1150:
1146:
1145:
1129:
1112:
1110:
1109:
1104:
1098:
1094:
1092:
1088:
1087:
1071:
1052:
1048:
1047:
1031:
1007:
991:
989:
988:
983:
978:
977:
959:
958:
930:
918:
906:
894:
878:
876:
875:
870:
815:
808:
806:
805:
800:
785:
774:
766:
762:
760:
759:
754:
752:
751:
746:
717:
713:
686:
678:
668:
664:
660:
656:
611:
607:
588:
584:
569:
565:
563:
562:
557:
552:
551:
542:
541:
532:
531:
502:
498:
414:generalized sets
403:Richard Dedekind
304:, who described
288:coined the word
281:
277:
273:
269:
234:
218:
202:
176:
172:
168:
141:In the multiset
137:
133:
129:
114:In the multiset
110:
98:
94:
90:
74:
70:
21:
7313:
7312:
7308:
7307:
7306:
7304:
7303:
7302:
7283:
7282:
7281:
7274:
7261:
7260:
7256:
7240:
7239:
7235:
7221:
7220:
7216:
7202:
7201:
7197:
7169:
7168:
7164:
7150:
7149:
7145:
7138:
7117:
7116:
7112:
7082:
7081:
7077:
7060:
7053:
7052:
7048:
7026:
7025:
7021:
6985:
6984:
6980:
6956:
6955:
6951:
6927:
6926:
6922:
6909:
6908:
6904:
6876:
6875:
6871:
6864:
6851:
6850:
6846:
6839:
6821:
6820:
6816:
6809:
6792:
6791:
6787:
6777:
6776:
6772:
6763:
6759:
6737:
6736:
6732:
6717:10.2307/1968168
6700:
6699:
6695:
6677:
6676:
6669:
6656:
6655:
6651:
6638:
6637:
6633:
6623:
6622:
6618:
6603:
6602:
6598:
6588:
6587:
6583:
6569:
6568:
6561:
6551:
6550:
6546:
6528:
6527:
6514:
6484:
6483:
6479:
6472:
6448:
6447:
6438:
6431:
6410:
6409:
6405:
6393:on 2011-06-10.
6375:
6374:
6370:
6366:
6361:
6360:
6329:
6314:
6286:
6282:
6251:
6232:
6225:
6218:
6190:
6188:
6184:
6179:
6139:
6114:Rough multisets
6111:Fuzzy multisets
6098:
6096:Generalizations
6090:linear operator
6041:
6028:
6018:
6014:
6010:
5968:
5940:
5927:
5880:
5852:
5827:
5810:
5809:
5783:
5781:
5743:
5715:
5711:
5707:
5670:
5653:
5652:
5613:
5609:
5605:
5600:
5599:
5515:
5456:
5441:
5426:
5393:
5389:
5385:
5380:
5379:
5378:, or complex):
5371:
5367:
5364:
5355:
5347:
5308:
5304:
5300:
5295:
5294:
5265:
5246:
5235:
5234:
5232:polynomial ring
5223:
5219:
5177:
5173:
5169:
5164:
5163:
5132:
5119:
5101:
5073:
5069:
5065:
5039:
5038:
5031:
4984:
4977:
4973:
4969:
4940:
4930:
4926:
4922:
4891:
4887:
4883:
4878:
4877:
4832:
4825:
4821:
4817:
4812:
4811:
4807:
4803:
4799:possibilities.
4763:
4753:
4749:
4745:
4740:
4739:
4732:
4728:
4724:
4720:
4709:
4699:
4650:
4649:
4589:
4585:
4581:
4520:
4516:
4512:
4507:
4506:
4443:
4436:
4432:
4428:
4399:
4389:
4385:
4381:
4350:
4346:
4342:
4337:
4336:
4329:
4285:
4274:
4267:
4263:
4259:
4228:
4224:
4220:
4215:
4214:
4181:
4174:
4163:
4123:
4119:
4115:
4110:
4109:
4108:can be written
4105:
4101:
4085:
4084:
4062:
4045:
4034:
4033:
3957:
3888:
3877:
3876:
3754:
3637:
3626:
3625:
3601:
3580:
3572:
3546:
3538:
3510:
3506:
3502:
3493:
3492:
3444:
3437:
3414:
3397:
3390:
3384:
3383:
3379:
3375:
3371:
3367:
3363:
3356:
3349:
3342:
3335:
3331:
3327:
3323:
3247:
3236:
3232:
3228:
3224:
3213:
3209:
3201:
3197:
3193:
3189:
3182:
3175:
3171:
3145:
3130:
3104:
3098:
3097:
3073:
3058:
3025:
3021:
3017:
3012:
3011:
2997:
2993:
2968:
2909:
2875:
2849:
2816:
2809:
2777:
2773:
2769:
2764:
2763:
2708:
2701:
2700:
2696:
2692:
2691:" to resemble "
2688:
2684:
2640:
2636:
2632:
2626:
2625:
2622:multiset number
2613:
2609:
2606:
2558:
2554:
2550:
2526:
2520:
2519:
2517:
2515:
2505:
2419:
2397:
2369:
2364:
2363:
2359:
2355:
2351:
2283:
2261:
2239:
2234:
2233:
2229:
2225:
2221:
2163:
2141:
2113:
2108:
2107:
2103:
2099:
2095:
2029:
2007:
1979:
1974:
1973:
1969:
1965:
1961:
1898:
1876:
1871:
1870:
1860:
1856:
1848:
1819:
1814:
1813:
1792:
1787:
1786:
1782:
1778:
1774:
1762:is finite. The
1733:
1695:
1646:
1645:
1597:
1559:
1558:
1539:
1538:
1534:
1515:
1514:
1488:
1463:
1462:
1458:
1454:
1451:natural numbers
1442:
1439:
1400:
1395:
1394:
1390:
1386:
1378:
1369:
1358:
1355:
1345:
1336:
1260:
1255:
1254:
1224:
1216:
1215:
1171:
1137:
1115:
1114:
1079:
1039:
1022:
1018:
1013:
1012:
997:
996:, the multiset
969:
950:
936:
935:
920:
908:
896:
880:
822:
821:
813:
791:
790:
776:
772:
764:
741:
724:
723:
715:
703:
693:
684:
674: ×
670:
666:
662:
658:
648:
609:
605:
597:solutions of a
586:
582:
568:{2, 2, 2, 3, 5}
567:
543:
533:
523:
512:
511:
500:
496:
489:
473:natural numbers
410:Hassler Whitney
384:Marius Nizolius
346:
279:
275:
271:
243:
220:
204:
188:
174:
170:
142:
135:
131:
115:
100:
96:
92:
80:
72:
68:
35:
28:
23:
22:
15:
12:
11:
5:
7311:
7309:
7301:
7300:
7295:
7285:
7284:
7280:
7279:
7272:
7254:
7233:
7214:
7195:
7162:
7143:
7136:
7110:
7075:
7046:
7019:
6998:(1): 346–368.
6978:
6949:
6938:(4): 209–214.
6920:
6902:
6889:(3): 570–588.
6869:
6862:
6844:
6837:
6814:
6807:
6785:
6770:
6757:
6746:(2): 171–178.
6730:
6711:(3): 405–414.
6693:
6667:
6649:
6631:
6616:
6596:
6581:
6559:
6544:
6512:
6477:
6470:
6462:Addison Wesley
6436:
6429:
6403:
6367:
6365:
6362:
6359:
6358:
6351:
6350:
6344:
6343:
6308:
6307:
6301:
6300:
6245:
6244:
6240:
6239:
6212:
6211:
6205:
6204:
6181:
6180:
6178:
6175:
6174:
6173:
6172:at Wikiversity
6161:
6156:
6151:
6146:
6138:
6135:
6134:
6133:
6130:
6127:
6126:Soft multisets
6124:
6118:
6115:
6112:
6109:
6097:
6094:
6040:
6037:
5995:
5990:
5987:
5984:
5981:
5978:
5975:
5971:
5967:
5964:
5961:
5958:
5955:
5950:
5947:
5943:
5937:
5934:
5930:
5926:
5923:
5920:
5917:
5914:
5902:
5899:
5896:
5893:
5890:
5887:
5883:
5879:
5876:
5873:
5870:
5867:
5862:
5859:
5855:
5851:
5848:
5845:
5842:
5837:
5834:
5830:
5826:
5823:
5820:
5817:
5804:exponentiation
5755:
5750:
5746:
5741:
5732:
5727:
5724:
5719:
5710:
5704:
5699:
5696:
5693:
5689:
5685:
5680:
5677:
5673:
5669:
5666:
5663:
5660:
5639:
5630:
5625:
5622:
5617:
5608:
5585:
5582:
5573:
5569:
5566:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5453:
5447:
5444:
5437:
5434:
5429:
5423:
5419:
5410:
5405:
5402:
5397:
5388:
5363:
5360:
5334:
5325:
5320:
5317:
5312:
5303:
5280:
5277:
5272:
5268:
5264:
5261:
5258:
5253:
5249:
5245:
5242:
5228:Hilbert series
5203:
5194:
5189:
5186:
5181:
5172:
5147:
5139:
5135:
5131:
5128:
5125:
5122:
5118:
5113:
5108:
5104:
5099:
5090:
5085:
5082:
5077:
5068:
5062:
5057:
5054:
5051:
5047:
5030:
5027:
5015:
5011:
5002:
4997:
4993:
4990:
4987:
4981:
4972:
4968:
4964:
4955:
4949:
4946:
4943:
4939:
4934:
4925:
4921:
4917:
4908:
4903:
4900:
4895:
4886:
4863:
4859:
4850:
4845:
4841:
4838:
4835:
4829:
4820:
4787:
4778:
4772:
4769:
4766:
4762:
4757:
4748:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4635:
4632:
4629:
4625:
4622:
4619:
4615:
4606:
4601:
4598:
4593:
4584:
4571:
4567:
4563:
4560:
4556:
4553:
4550:
4546:
4537:
4532:
4529:
4524:
4515:
4494:
4491:
4488:
4485:
4482:
4470:
4461:
4456:
4452:
4449:
4446:
4440:
4431:
4427:
4423:
4414:
4408:
4405:
4402:
4398:
4393:
4384:
4380:
4376:
4367:
4362:
4359:
4354:
4345:
4328:
4325:
4313:
4309:
4300:
4294:
4291:
4288:
4283:
4280:
4277:
4271:
4262:
4258:
4254:
4245:
4240:
4237:
4232:
4223:
4213:Additionally,
4202:
4196:
4191:
4187:
4184:
4178:
4170:
4166:
4162:
4159:
4156:
4153:
4149:
4140:
4135:
4132:
4127:
4118:
4083:
4077:
4074:
4071:
4068:
4065:
4060:
4057:
4054:
4051:
4048:
4042:
4039:
4037:
4035:
4032:
4024:
4021:
4018:
4015:
4012:
4008:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3954:
3951:
3948:
3945:
3942:
3938:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3885:
3882:
3880:
3878:
3875:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3774:
3770:
3767:
3764:
3761:
3758:
3751:
3748:
3745:
3742:
3739:
3735:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3634:
3631:
3629:
3627:
3624:
3618:
3613:
3610:
3605:
3599:
3593:
3590:
3586:
3583:
3578:
3575:
3569:
3563:
3558:
3555:
3550:
3544:
3541:
3539:
3536:
3527:
3522:
3519:
3514:
3505:
3501:
3500:
3480:
3477:
3474:
3468:
3463:
3459:
3456:
3453:
3450:
3447:
3441:
3435:
3429:
3423:
3420:
3417:
3412:
3409:
3406:
3403:
3400:
3394:
3346:
3345:
3157:
3151:
3148:
3141:
3138:
3133:
3127:
3121:
3116:
3113:
3108:
3085:
3079:
3076:
3069:
3066:
3061:
3055:
3051:
3042:
3037:
3034:
3029:
3020:
2980:
2974:
2971:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2906:
2900:
2897:
2894:
2891:
2888:
2885:
2881:
2878:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2846:
2840:
2835:
2831:
2828:
2825:
2822:
2819:
2813:
2807:
2803:
2794:
2789:
2786:
2781:
2772:
2732:
2725:
2720:
2717:
2712:
2666:
2657:
2652:
2649:
2644:
2635:
2588:
2584:
2575:
2570:
2567:
2562:
2553:
2549:
2543:
2538:
2535:
2530:
2504:
2501:
2474:
2473:
2462:
2459:
2456:
2453:
2450:
2446:
2443:
2440:
2437:
2434:
2431:
2426:
2422:
2418:
2415:
2412:
2409:
2404:
2400:
2396:
2393:
2390:
2387:
2384:
2381:
2376:
2372:
2341:
2330:disjoint union
2317:
2314:
2311:
2308:
2305:
2301:
2298:
2295:
2290:
2286:
2282:
2279:
2276:
2273:
2268:
2264:
2260:
2257:
2254:
2251:
2246:
2242:
2211:
2200:
2197:
2194:
2191:
2188:
2184:
2181:
2178:
2175:
2170:
2166:
2162:
2159:
2156:
2153:
2148:
2144:
2140:
2137:
2134:
2131:
2128:
2125:
2120:
2116:
2077:
2066:
2063:
2060:
2057:
2054:
2050:
2047:
2044:
2041:
2036:
2032:
2028:
2025:
2022:
2019:
2014:
2010:
2006:
2003:
2000:
1997:
1994:
1991:
1986:
1982:
1943:
1932:
1929:
1926:
1923:
1920:
1916:
1913:
1910:
1905:
1901:
1897:
1894:
1891:
1888:
1883:
1879:
1831:
1826:
1822:
1799:
1795:
1764:empty multiset
1751:
1748:
1745:
1740:
1736:
1730:
1727:
1724:
1720:
1716:
1713:
1710:
1707:
1702:
1698:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1670:
1666:
1662:
1658:
1654:
1640:A multiset is
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1604:
1600:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1546:
1533:in a universe
1522:
1513:of a multiset
1471:
1438:
1435:
1415:
1412:
1407:
1403:
1374:
1347:
1341:
1334:indexed family
1300:indeterminates
1275:
1272:
1267:
1263:
1242:
1239:
1236:
1231:
1227:
1223:
1200:
1199:
1188:
1183:
1178:
1174:
1170:
1167:
1162:
1158:
1154:
1149:
1144:
1140:
1136:
1133:
1128:
1124:
1101:
1097:
1091:
1086:
1082:
1078:
1075:
1070:
1066:
1062:
1059:
1056:
1051:
1046:
1042:
1038:
1035:
1030:
1026:
1021:
981:
976:
972:
968:
965:
962:
957:
953:
949:
946:
943:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
798:
750:
745:
740:
737:
734:
731:
720:underlying set
692:
689:
555:
550:
546:
540:
536:
530:
526:
522:
519:
488:
485:
455:that respects
416:("sets" whose
380:Bhāskarāchārya
345:
342:
302:Bhāskarāchārya
179:
178:
139:
130:, the element
112:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7310:
7299:
7296:
7294:
7291:
7290:
7288:
7275:
7269:
7265:
7258:
7255:
7249:
7244:
7237:
7234:
7229:
7225:
7218:
7215:
7210:
7206:
7199:
7196:
7190:
7185:
7181:
7177:
7173:
7166:
7163:
7158:
7154:
7147:
7144:
7139:
7133:
7129:
7125:
7121:
7114:
7111:
7105:
7100:
7096:
7092:
7091:
7086:
7079:
7076:
7071:
7065:
7057:
7050:
7047:
7042:
7038:
7034:
7030:
7023:
7020:
7015:
7011:
7006:
7001:
6997:
6993:
6989:
6982:
6979:
6974:
6970:
6966:
6962:
6961:
6953:
6950:
6945:
6941:
6937:
6933:
6932:
6924:
6921:
6916:
6912:
6906:
6903:
6897:
6892:
6888:
6884:
6880:
6873:
6870:
6865:
6863:0-521-56069-1
6859:
6855:
6848:
6845:
6840:
6838:0-521-55309-1
6834:
6830:
6829:
6824:
6818:
6815:
6810:
6804:
6799:
6798:
6789:
6786:
6781:
6774:
6771:
6767:
6761:
6758:
6753:
6749:
6745:
6741:
6734:
6731:
6726:
6722:
6718:
6714:
6710:
6706:
6705:
6697:
6694:
6689:
6685:
6681:
6674:
6672:
6668:
6663:
6659:
6653:
6650:
6645:
6641:
6635:
6632:
6627:
6620:
6617:
6612:
6611:
6606:
6600:
6597:
6593:(6): 319–322.
6592:
6585:
6582:
6577:
6573:
6566:
6564:
6560:
6555:
6548:
6545:
6541:(4): 319–352.
6540:
6536:
6532:
6525:
6523:
6521:
6519:
6517:
6513:
6507:
6502:
6498:
6494:
6493:
6488:
6481:
6478:
6473:
6471:0-201-89684-2
6467:
6463:
6459:
6455:
6451:
6445:
6443:
6441:
6437:
6432:
6430:0-7637-2210-3
6426:
6422:
6417:
6416:
6407:
6404:
6400:
6398:
6392:
6388:
6387:
6382:
6378:
6377:Cantor, Georg
6372:
6369:
6363:
6349:
6346:
6345:
6341:
6337:
6334:
6333:
6327:
6321:
6317:
6306:
6303:
6302:
6298:
6294:
6291:
6290:
6280:
6279:empty product
6274:
6270:
6266:
6262:
6258:
6254:
6242:
6241:
6237:
6236:
6228:
6221:
6210:
6207:
6206:
6202:
6198:
6195:
6194:
6186:
6183:
6176:
6171:
6166:
6162:
6160:
6157:
6155:
6152:
6150:
6147:
6144:
6141:
6140:
6136:
6131:
6128:
6125:
6123:
6122:step function
6119:
6116:
6113:
6110:
6107:
6103:
6102:
6101:
6095:
6093:
6091:
6087:
6083:
6079:
6074:
6069:
6067:
6063:
6058:
6054:
6050:
6046:
6045:combinatorics
6038:
6036:
6034:
6025:
6021:
6007:
5993:
5985:
5982:
5979:
5973:
5965:
5962:
5959:
5953:
5948:
5945:
5935:
5932:
5924:
5921:
5918:
5897:
5894:
5891:
5885:
5877:
5874:
5871:
5865:
5860:
5857:
5849:
5846:
5843:
5835:
5832:
5824:
5821:
5818:
5807:
5805:
5801:
5797:
5793:
5788:| < 1
5786:
5779:
5775:
5771:
5770:Taylor series
5766:
5753:
5748:
5744:
5739:
5725:
5722:
5708:
5697:
5694:
5691:
5687:
5683:
5678:
5675:
5667:
5664:
5661:
5637:
5623:
5620:
5606:
5596:
5583:
5580:
5567:
5564:
5552:
5549:
5543:
5540:
5537:
5528:
5525:
5522:
5516:
5508:
5505:
5502:
5499:
5496:
5490:
5484:
5481:
5478:
5469:
5466:
5463:
5457:
5451:
5445:
5442:
5432:
5427:
5421:
5417:
5403:
5400:
5386:
5377:
5361:
5359:
5353:
5332:
5318:
5315:
5301:
5291:
5278:
5270:
5266:
5262:
5259:
5256:
5251:
5247:
5240:
5233:
5229:
5218:
5201:
5187:
5184:
5170:
5161:
5145:
5137:
5129:
5126:
5123:
5116:
5111:
5106:
5102:
5097:
5083:
5080:
5066:
5055:
5052:
5049:
5045:
5036:
5028:
5026:
5013:
5009:
4995:
4991:
4988:
4985:
4970:
4966:
4962:
4947:
4944:
4941:
4937:
4923:
4919:
4915:
4901:
4898:
4884:
4874:
4861:
4857:
4843:
4839:
4836:
4833:
4818:
4800:
4785:
4770:
4767:
4764:
4760:
4746:
4735:
4716:
4712:
4706:
4702:
4682:
4679:
4676:
4673:
4670:
4664:
4658:
4646:
4633:
4630:
4627:
4623:
4620:
4617:
4613:
4599:
4596:
4582:
4569:
4561:
4558:
4554:
4551:
4548:
4544:
4530:
4527:
4513:
4492:
4489:
4486:
4483:
4480:
4468:
4454:
4450:
4447:
4444:
4429:
4425:
4421:
4406:
4403:
4400:
4396:
4382:
4378:
4374:
4360:
4357:
4343:
4334:
4326:
4324:
4311:
4307:
4292:
4289:
4286:
4281:
4278:
4275:
4260:
4256:
4252:
4238:
4235:
4221:
4200:
4189:
4185:
4182:
4168:
4160:
4157:
4151:
4147:
4133:
4130:
4116:
4098:
4081:
4075:
4072:
4069:
4066:
4063:
4058:
4055:
4052:
4049:
4046:
4040:
4038:
4030:
4019:
4013:
4006:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3949:
3943:
3936:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3883:
3881:
3873:
3860:
3854:
3848:
3842:
3836:
3830:
3824:
3818:
3812:
3806:
3800:
3794:
3788:
3782:
3772:
3765:
3759:
3746:
3740:
3733:
3723:
3717:
3711:
3705:
3699:
3693:
3687:
3681:
3675:
3669:
3663:
3657:
3651:
3645:
3632:
3630:
3622:
3611:
3608:
3597:
3591:
3588:
3584:
3581:
3576:
3573:
3567:
3556:
3553:
3542:
3540:
3534:
3520:
3517:
3503:
3478:
3475:
3472:
3461:
3457:
3454:
3451:
3448:
3445:
3433:
3421:
3418:
3415:
3410:
3407:
3404:
3401:
3398:
3359:
3352:
3341:
3340:
3339:
3319:
3315:
3311:
3307:
3303:
3299:
3295:
3291:
3287:
3283:
3279:
3275:
3271:
3267:
3263:
3259:
3255:
3251:
3245:
3240:
3220:
3216:
3207:
3185:
3178:
3168:
3155:
3149:
3146:
3139:
3136:
3131:
3125:
3114:
3111:
3083:
3077:
3074:
3064:
3059:
3053:
3049:
3035:
3032:
3018:
3010:
3004:
3000:
2978:
2972:
2969:
2961:
2958:
2955:
2952:
2949:
2943:
2937:
2934:
2931:
2922:
2919:
2916:
2910:
2904:
2898:
2892:
2889:
2886:
2879:
2876:
2871:
2865:
2862:
2859:
2856:
2853:
2844:
2833:
2829:
2826:
2823:
2820:
2817:
2805:
2801:
2787:
2784:
2770:
2760:
2758:
2754:
2750:
2746:
2730:
2718:
2715:
2682:
2664:
2650:
2647:
2633:
2623:
2619:
2605:
2586:
2582:
2568:
2565:
2551:
2547:
2536:
2533:
2513:
2509:
2502:
2500:
2498:
2494:
2490:
2485:
2483:
2482:disjoint sets
2479:
2460:
2457:
2454:
2451:
2441:
2438:
2432:
2424:
2420:
2416:
2410:
2402:
2398:
2388:
2382:
2374:
2370:
2349:
2345:
2342:
2339:
2335:
2331:
2315:
2312:
2309:
2306:
2296:
2288:
2284:
2280:
2274:
2266:
2262:
2258:
2252:
2244:
2240:
2219:
2215:
2212:
2198:
2195:
2192:
2189:
2176:
2168:
2164:
2160:
2154:
2146:
2142:
2132:
2126:
2118:
2114:
2093:
2089:
2085:
2081:
2080:Intersection:
2078:
2064:
2061:
2058:
2055:
2042:
2034:
2030:
2026:
2020:
2012:
2008:
1998:
1992:
1984:
1980:
1959:
1955:
1951:
1947:
1944:
1930:
1927:
1924:
1921:
1911:
1903:
1899:
1895:
1889:
1881:
1877:
1867:
1863:
1854:
1847:
1844:
1843:
1842:
1829:
1824:
1820:
1797:
1793:
1771:
1769:
1765:
1746:
1738:
1734:
1728:
1725:
1722:
1718:
1714:
1708:
1700:
1696:
1687:
1681:
1678:
1675:
1672:
1668:
1664:
1656:
1643:
1638:
1625:
1619:
1616:
1610:
1602:
1598:
1594:
1591:
1588:
1585:
1579:
1573:
1567:
1564:
1544:
1520:
1512:
1507:
1505:
1501:
1497:
1492:
1486:
1452:
1448:
1436:
1434:
1432:
1427:
1413:
1410:
1405:
1401:
1384:
1377:
1373:
1367:
1364:
1354:
1350:
1344:
1340:
1335:
1330:
1328:
1324:
1320:
1316:
1312:
1308:
1305:
1301:
1297:
1293:
1289:
1273:
1270:
1265:
1261:
1237:
1234:
1229:
1225:
1213:
1209:
1205:
1186:
1176:
1172:
1165:
1160:
1156:
1152:
1142:
1138:
1131:
1126:
1122:
1099:
1095:
1084:
1080:
1073:
1068:
1064:
1060:
1057:
1054:
1044:
1040:
1033:
1028:
1024:
1019:
1011:
1010:
1009:
1005:
1001:
995:
974:
970:
966:
963:
960:
955:
951:
944:
941:
932:
928:
924:
916:
912:
904:
900:
892:
888:
884:
863:
860:
857:
854:
845:
839:
836:
833:
819:
810:
787:
783:
779:
770:
748:
735:
732:
729:
721:
711:
707:
702:
698:
690:
688:
682:
677:
673:
655:
651:
646:
642:
638:
634:
630:
626:
622:
618:
613:
604:
600:
596:
592:
580:
576:
571:
553:
548:
544:
538:
534:
528:
524:
520:
517:
510:
506:
494:
493:prime factors
486:
484:
482:
478:
475:, giving the
474:
470:
466:
462:
458:
454:
450:
446:
442:
438:
435:, defining a
434:
430:
427:
423:
420:may take any
419:
415:
411:
406:
404:
399:
397:
393:
389:
385:
381:
376:
374:
373:Peter Deutsch
370:
366:
361:
359:
355:
351:
343:
341:
339:
335:
331:
327:
323:
319:
315:
311:
307:
303:
299:
295:
291:
287:
283:
267:
263:
259:
255:
251:
247:
241:
236:
232:
228:
224:
216:
212:
208:
200:
196:
192:
186:
185:
166:
162:
158:
154:
150:
146:
140:
127:
123:
119:
113:
108:
104:
88:
84:
78:
77:
76:
66:
65:
60:
56:
52:
48:
44:
40:
33:
19:
7263:
7257:
7248:math/0403186
7236:
7227:
7223:
7217:
7208:
7198:
7179:
7175:
7165:
7156:
7152:
7146:
7119:
7113:
7094:
7088:
7078:
7055:
7049:
7032:
7028:
7022:
6995:
6991:
6981:
6964:
6958:
6952:
6935:
6929:
6923:
6914:
6905:
6886:
6882:
6872:
6853:
6847:
6827:
6817:
6796:
6788:
6779:
6773:
6765:
6760:
6743:
6739:
6733:
6708:
6702:
6696:
6679:
6661:
6652:
6643:
6640:Wallis, John
6634:
6625:
6619:
6609:
6599:
6590:
6584:
6575:
6571:
6553:
6547:
6538:
6535:Modern Logic
6534:
6499:(1): 36–66.
6496:
6490:
6480:
6453:
6414:
6406:
6396:
6394:
6391:the original
6384:
6371:
6347:
6339:
6335:
6319:
6315:
6304:
6296:
6292:
6272:
6268:
6264:
6260:
6256:
6252:
6226:
6219:
6208:
6200:
6196:
6189:The formula
6185:
6099:
6081:
6077:
6073:Richard Rado
6070:
6052:
6042:
6039:Applications
6023:
6019:
6008:
5808:
5799:
5784:
5777:
5773:
5767:
5597:
5365:
5292:
5032:
4875:
4801:
4733:
4714:
4710:
4707:
4700:
4647:
4330:
4099:
3357:
3350:
3347:
3317:
3313:
3309:
3305:
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3257:
3253:
3249:
3241:
3218:
3214:
3210:{1, 2, 3, 4}
3205:
3183:
3176:
3169:
3002:
2998:
2761:
2687:multichoose
2621:
2617:
2607:
2486:
2477:
2475:
2347:
2343:
2217:
2213:
2091:
2087:
2084:intersection
2083:
2079:
1957:
1953:
1949:
1945:
1865:
1861:
1852:
1845:
1772:
1763:
1641:
1639:
1510:
1508:
1495:
1493:
1446:
1440:
1428:
1375:
1371:
1365:
1352:
1348:
1342:
1338:
1331:
1326:
1322:
1318:
1314:
1310:
1306:
1303:
1211:
1207:
1203:
1201:
1003:
999:
933:
926:
922:
914:
910:
902:
898:
890:
886:
882:
811:
788:
781:
777:
769:multiplicity
768:
719:
709:
705:
701:ordered pair
696:
694:
675:
671:
653:
649:
614:
572:
490:
480:
477:multiplicity
476:
468:
464:
460:
456:
448:
444:
436:
428:
413:
407:
400:
392:Jean Prestet
377:
368:
362:
353:
349:
347:
337:
333:
330:weighted set
329:
325:
321:
317:
313:
309:
297:
294:Donald Knuth
289:
284:
265:
261:
257:
253:
249:
245:
237:
230:
226:
222:
214:
210:
206:
198:
194:
190:
182:
180:
164:
160:
156:
152:
148:
144:
125:
121:
117:
106:
102:
86:
82:
64:multiplicity
63:
50:
46:
42:
36:
6578:(2): 73–92.
6271: −1)/
6117:Hybrid sets
6106:real number
6076:polynomial
6062:multigraphs
5374:(negative,
2344:Difference:
1461:to the set
617:eigenvalues
479:of element
461:multinumber
396:John Wallis
358:tally marks
240:cardinality
39:mathematics
7287:Categories
7230:: 165–167.
6911:Libkin, L.
6768:, Pearson.
6364:References
6285:. However
6154:Set theory
6149:Quasi-sets
5806:, notably
3382:. This is
3380:18 + 4 − 1
3376:18 + 4 − 1
3372:18 + 4 − 1
3364:18 + 4 − 1
3353:= 18
3223:), namely
3188:), namely
2602:See also:
2348:difference
1859:, denoted
1846:Inclusion:
1393:such that
1294:. Also, a
994:finite set
691:Definition
679:matrix in
635:, and the
334:collection
7097:: 64–74.
7064:cite book
7035:: 23–37.
6967:: 77–97.
6342: −1
6328:; indeed
6299: −1
6203: −1
6084:) or the
5986:β
5983:α
5980:−
5974:−
5963:−
5949:β
5946:−
5936:α
5933:−
5922:−
5898:β
5892:α
5886:−
5875:−
5861:β
5858:−
5847:−
5836:α
5833:−
5822:−
5723:α
5703:∞
5688:∑
5679:α
5676:−
5665:−
5621:α
5581:α
5568:∈
5561:for
5550:⋯
5541:−
5526:−
5506:−
5497:α
5491:⋯
5479:α
5464:α
5458:α
5436:¯
5428:α
5401:α
5354:value of
5260:…
5160:monomials
5127:−
5061:∞
5046:∑
4989:−
4945:−
4837:−
4768:−
4736:− 1
4677:…
4562:∈
4476:for
4448:−
4404:−
4290:−
4183:−
4158:−
4073:⋅
4067:⋅
4056:⋅
4050:⋅
4020:⋅
4014:⋅
4007:⋅
3999:⋅
3993:⋅
3987:⋯
3981:⋅
3975:⋅
3969:⋅
3963:⋅
3950:⋅
3944:⋅
3937:⋅
3929:⋅
3923:⋅
3917:⋯
3911:⋅
3905:⋅
3899:⋅
3893:⋅
3861:⋅
3855:⋅
3849:⋅
3843:⋅
3837:⋅
3831:⋅
3825:⋅
3819:⋅
3813:⋅
3807:⋅
3801:⋅
3795:⋅
3789:⋅
3783:⋅
3773:⋅
3766:⋅
3760:⋅
3747:⋅
3741:⋅
3734:⋅
3724:⋅
3718:⋅
3712:⋅
3706:⋅
3700:⋅
3694:⋅
3688:⋅
3682:⋅
3676:⋅
3670:⋅
3664:⋅
3658:⋅
3652:⋅
3646:⋅
3455:−
3419:−
3408:−
3237:{2, 3, 4}
3233:{1, 3, 4}
3229:{1, 2, 4}
3225:{1, 2, 3}
3221:− 1
3202:{2, 2, 2}
3198:{1, 2, 2}
3194:{1, 1, 2}
3190:{1, 1, 1}
3140:_
3068:¯
3005:− 1
2959:−
2944:⋯
2890:−
2863:−
2827:−
2743:Like the
2512:Bijection
2455:∈
2449:∀
2417:−
2310:∈
2304:∀
2193:∈
2187:∀
2059:∈
2053:∀
1925:∈
1919:∀
1896:≤
1726:∈
1719:∑
1682:
1676:∈
1669:∑
1595:∣
1589:∈
1568:
1431:cardinals
1363:index set
1153:⋯
1058:…
964:…
861:∈
797:∞
739:→
733::
641:dimension
447:", and a
433:morphisms
356:strokes,
18:Multisets
7182:: 1–20.
7014:42766971
6825:(1997).
6660:(1888).
6642:(1685).
6607:(1650).
6452:(1998).
6397:separate
6283:1/0! = 1
6137:See also
6086:spectrum
6080: (
6066:vertices
5792:identity
2478:disjoint
1853:included
1447:universe
1357:, where
1296:monomial
697:multiset
507:has the
487:Examples
467: (
453:function
449:morphism
437:multiset
426:category
298:multiset
290:multiset
79:The set
59:elements
43:multiset
6725:1968168
6281:giving
6263:+2)...(
5352:complex
5230:of the
3206:subsets
2695:choose
2088:infimum
1954:maximum
1511:support
925:, 1), (
901:, 2), (
816:by its
718:is the
657:(where
643:of the
627:of the
595:complex
422:integer
344:History
7270:
7134:
7012:
6860:
6835:
6805:
6723:
6468:
6427:
6354: )
6330:(
6311: )
6287:(
6248: )
6233:(
6215: )
6191:(
6022:> −
5782:|
5217:degree
4876:Thus,
4717:> 0
3172:{1, 2}
2699:" for
1946:Union:
1642:finite
1504:subset
714:where
645:kernel
621:matrix
603:degree
587:{4, 4}
583:{3, 5}
336:, and
326:sample
278:, and
184:tuples
7243:arXiv
7010:S2CID
6721:JSTOR
6423:–30.
6177:Notes
6088:of a
5780:with
5768:This
4505:with
3368:4 − 1
3334:s, 7
3330:s, 3
3326:s, 2
3244:prove
2094:) of
1960:) of
1950:union
1869:, if
1768:empty
1502:of a
1383:image
992:is a
818:graph
669:be a
625:roots
619:of a
457:sorts
369:bags,
338:suite
314:bunch
49:, or
7268:ISBN
7180:2012
7132:ISBN
7070:link
6858:ISBN
6833:ISBN
6803:ISBN
6466:ISBN
6425:ISBN
6259:+1)(
5776:and
5376:real
5033:The
4631:>
4490:>
3476:1330
2497:even
2354:and
2346:the
2224:and
2216:the
2214:Sum:
2098:and
2082:the
1964:and
1948:the
1812:and
1777:and
1679:Supp
1617:>
1565:Supp
1509:The
929:, 1)
905:, 1)
577:. A
445:sort
322:heap
310:list
238:The
203:and
173:and
95:and
71:and
51:mset
45:(or
41:, a
7184:doi
7124:doi
7099:doi
7037:doi
7000:doi
6969:doi
6940:doi
6891:doi
6748:doi
6713:doi
6684:doi
6501:doi
6322:= 0
6229:= 0
6222:= 0
6092:."
6057:SQL
6053:bag
6009:If
5908:and
5798:in
5794:of
5293:As
5222:in
4802:If
4703:= 0
4576:and
3360:= 4
3322:(6
3186:= 3
3179:= 2
2620:or
2493:odd
2392:max
2350:of
2220:of
2218:sum
2136:min
2090:or
2002:max
1956:or
1855:in
1851:is
1329:}.
1253:or
934:If
919:as
895:as
647:of
601:of
518:120
505:120
429:Mul
318:bag
55:set
47:bag
37:In
7289::
7226:.
7207:.
7178:.
7174:.
7155:.
7130:.
7095:91
7093:.
7087:.
7066:}}
7062:{{
7033:13
7031:.
7008:.
6996:31
6994:.
6990:.
6965:33
6963:.
6936:56
6934:.
6887:52
6885:.
6881:.
6744:33
6742:.
6719:.
6709:34
6707:.
6670:^
6576:37
6574:.
6562:^
6537:.
6533:.
6515:^
6497:30
6495:.
6489:.
6464:.
6456:.
6439:^
6421:29
6318:=
6238:−1
5358:.
5162:,
4713:,
4665::=
4634:0.
4331:A
4059:21
4053:20
4047:19
4002:18
3996:17
3990:16
3953:21
3947:20
3941:19
3932:18
3926:17
3920:16
3864:18
3858:17
3852:16
3846:15
3840:14
3834:13
3828:12
3822:11
3816:10
3750:21
3744:20
3738:19
3727:18
3721:17
3715:16
3709:15
3703:14
3697:13
3691:12
3685:11
3679:10
3609:21
3582:18
3574:21
3557:18
3554:21
3521:18
3462:18
3452:18
3405:18
3316:,
3312:,
3308:,
3304:,
3300:,
3296:,
3292:,
3288:,
3284:,
3280:,
3276:,
3272:,
3268:,
3264:,
3260:,
3256:,
3252:,
3239:.
3235:,
3231:,
3227:,
3217:+
3200:,
3196:,
3192:,
3181:,
3001:+
2759:.
1864:⊆
1580::=
1491:.
1325:,
1321:,
1317:,
1313:,
1210:,
1206:,
1002:,
921:{(
913:,
897:{(
889:,
885:,
786:.
708:,
695:A
654:λI
652:−
612:.
570:.
405:.
365:AI
340:.
332:,
328:,
324:,
320:,
316:,
312:,
274:,
264:,
260:,
256:,
252:,
248:,
229:,
225:,
213:,
209:,
197:,
193:,
169:,
163:,
159:,
155:,
151:,
147:,
124:,
120:,
105:,
85:,
7276:.
7251:.
7245::
7228:3
7192:.
7186::
7157:5
7140:.
7126::
7107:.
7101::
7072:)
7043:.
7039::
7016:.
7002::
6975:.
6971::
6946:.
6942::
6899:.
6893::
6866:.
6841:.
6811:.
6754:.
6750::
6727:.
6715::
6690:.
6686::
6539:1
6509:.
6503::
6474:.
6433:.
6348:k
6340:k
6338:+
6336:n
6320:k
6316:n
6305:k
6297:k
6295:+
6293:n
6275:!
6273:k
6269:k
6267:+
6265:n
6261:n
6257:n
6255:(
6253:n
6243:0
6227:k
6220:n
6209:k
6201:k
6199:+
6197:n
6108:)
6082:x
6078:f
6029:α
6024:n
6020:k
6015:n
6011:α
5994:,
5989:)
5977:(
5970:)
5966:X
5960:1
5957:(
5954:=
5942:)
5929:)
5925:X
5919:1
5916:(
5913:(
5901:)
5895:+
5889:(
5882:)
5878:X
5872:1
5869:(
5866:=
5854:)
5850:X
5844:1
5841:(
5829:)
5825:X
5819:1
5816:(
5800:X
5785:X
5778:X
5774:α
5754:.
5749:k
5745:X
5740:)
5731:)
5726:k
5718:(
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5698:0
5695:=
5692:k
5684:=
5672:)
5668:X
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5659:(
5638:)
5629:)
5624:k
5616:(
5607:(
5584:.
5572:N
5565:k
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5488:)
5485:2
5482:+
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5473:)
5470:1
5467:+
5461:(
5452:=
5446:!
5443:k
5433:k
5422:=
5418:)
5409:)
5404:k
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5387:(
5372:α
5368:n
5356:n
5348:n
5333:)
5324:)
5319:d
5316:n
5311:(
5302:(
5279:.
5276:]
5271:n
5267:x
5263:,
5257:,
5252:1
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5244:[
5241:k
5224:n
5220:d
5202:)
5193:)
5188:d
5185:n
5180:(
5171:(
5146:.
5138:n
5134:)
5130:t
5124:1
5121:(
5117:1
5112:=
5107:d
5103:t
5098:)
5089:)
5084:d
5081:n
5076:(
5067:(
5056:0
5053:=
5050:d
5014:.
5010:)
5001:)
4996:k
4992:1
4986:n
4980:(
4971:(
4967:+
4963:)
4954:)
4948:1
4942:k
4938:n
4933:(
4924:(
4920:=
4916:)
4907:)
4902:k
4899:n
4894:(
4885:(
4862:.
4858:)
4849:)
4844:k
4840:1
4834:n
4828:(
4819:(
4808:k
4804:n
4786:)
4777:)
4771:1
4765:k
4761:n
4756:(
4747:(
4734:k
4729:n
4725:n
4721:k
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4686:}
4683:n
4680:,
4674:,
4671:1
4668:{
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4659:n
4656:[
4628:k
4624:,
4621:0
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4614:)
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4597:0
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4583:(
4570:,
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4555:,
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4549:=
4545:)
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4531:0
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4460:)
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4430:(
4426:+
4422:)
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4392:(
4383:(
4379:=
4375:)
4366:)
4361:k
4358:n
4353:(
4344:(
4312:.
4308:)
4299:)
4293:1
4287:n
4282:1
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4276:k
4270:(
4261:(
4257:=
4253:)
4244:)
4239:k
4236:n
4231:(
4222:(
4201:.
4195:)
4190:k
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4177:(
4169:k
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4155:(
4152:=
4148:)
4139:)
4134:k
4131:n
4126:(
4117:(
4106:n
4102:k
4082:.
4076:3
4070:2
4064:1
4041:=
4031:,
4023:3
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4011:1
3984:5
3978:4
3972:3
3966:2
3960:1
3914:5
3908:4
3902:3
3896:2
3890:1
3884:=
3874:,
3810:9
3804:8
3798:7
3792:6
3786:5
3780:4
3769:3
3763:2
3757:1
3673:9
3667:8
3661:7
3655:6
3649:5
3643:4
3633:=
3623:,
3617:)
3612:3
3604:(
3598:=
3592:!
3589:3
3585:!
3577:!
3568:=
3562:)
3549:(
3543:=
3535:)
3526:)
3518:4
3513:(
3504:(
3479:,
3473:=
3467:)
3458:1
3449:+
3446:4
3440:(
3434:=
3428:)
3422:1
3416:4
3411:1
3402:+
3399:4
3393:(
3358:n
3351:k
3336:d
3332:c
3328:b
3324:a
3320:}
3318:d
3314:d
3310:d
3306:d
3302:d
3298:d
3294:d
3290:c
3286:c
3282:c
3278:b
3274:b
3270:a
3266:a
3262:a
3258:a
3254:a
3250:a
3248:{
3219:k
3215:n
3184:k
3177:n
3156:.
3150:!
3147:k
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3126:=
3120:)
3115:k
3112:n
3107:(
3084:,
3078:!
3075:k
3065:k
3060:n
3054:=
3050:)
3041:)
3036:k
3033:n
3028:(
3019:(
3003:k
2999:n
2994:k
2979:,
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2941:)
2938:2
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2869:)
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2834:k
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2806:=
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2731:.
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2548:=
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2421:m
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2408:(
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2395:(
2389:=
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2383:x
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2259:=
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2253:x
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2199:.
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2169:B
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2018:(
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2005:(
1999:=
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1626:.
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909:{
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864:A
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855::
852:)
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704:(
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663:A
659:λ
650:A
610:d
606:d
554:,
549:1
545:5
539:1
535:3
529:3
525:2
521:=
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481:x
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354:n
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280:c
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221:{
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205:{
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191:a
189:{
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171:a
167:}
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157:b
153:a
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145:a
143:{
136:b
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128:}
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116:{
109:}
107:b
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101:{
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89:}
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83:a
81:{
73:b
69:a
34:.
20:)
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