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### An asymptotic formula for the number of classes of sets of n indistinguishable elements

James F Lynch
1975 Journal of combinatorial theory. Series A
Ulam for originally suggesting this problem and to Professor Jan Mycielski for his helpful suggestions during the preparation of this note .  ...  1 This note addresses the following problem posed by S . M . Ulam . Find a concise expression for the number of classes of sets of n indistinguishable elements .  ...  Here, a class of sets is a set whose members are subsets of a given finite set of n elements, and referring to the elements as indistinguishable means that two classes are identical if some permutation  ...

### V. N. Sachkov Combinatorial methods in discrete mathematics (Encyclopedia of Mathematics and its Applications, Vol. 55, Cambridge University Press, Cambridge, 1996), xiii + 306 pp., 0 521 45513 8 (hardback), £45 (US\$69.95)

I. Anderson
1997 Proceedings of the Edinburgh Mathematical Society
A number of examples are given; for example, asymptotic formulae for Stirling numbers of the second kind and the Hardy-Ramanujan asymptotic formula for the number of partitions of an integer are presented  ...  Enumeration problems are viewed as concerning mappings from a set X to a set Y with permutation groups G and H on X and Y which give equivalence classes of elements, equivalent elements being regarded  ...

### Page 4849 of Mathematical Reviews Vol. , Issue 89I [page]

1989 Mathematical Reviews
Combinatorially it should be noted that n”~'Q(n) is the number of connected mappings of an n-element set into itself.  ...  Thus, from a combinatorial point of view, f,,(y) is the generating polynomial for endofunctions of an n-element set with a weight y put on each cycle (or connected component).  ...

### Counting self-dual interval orders

Vít Jelínek
2011 Discrete Mathematics & Theoretical Computer Science
, including their size, magnitude, and the number of minimal and maximal elements.  ...  In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics.  ...  They obtained asymptotic bounds for the number of these matrices, as well as recurrence formulas that allowed them to compute several exact initial values.  ...

### Page 1108 of Mathematical Reviews Vol. 51, Issue 4 [page]

1976 Mathematical Reviews
Lynch, James F. 7892 An asymptotic formula for the number of classes of sets of n indistinguishable elements. J. Combinatorial Theory Ser. A 19 (1975), 109-112.  ...  In the present note, the author deduces an asymptotic expression for the number N(n, d) of classes in the dth power set of an n element set.  ...

### (Pseudo) Random Quantum States with Binary Phase [article]

Zvika Brakerski, Omri Shmueli
2019 arXiv   pre-print
That is, any polynomial number of copies of the aforementioned state is within exponentially small trace distance from the same number of copies of a Haar random state.  ...  We observe that replacing the pseudorandom function with a (2t)-wise independent function (either in our construction or in previous work), results in an explicit construction for quantum state t-designs  ...  We also thank Aram Harrow for providing advice regarding the state of the art.  ...

### Counting general and self-dual interval orders

Vít Jelínek
2012 Journal of combinatorial theory. Series A
In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements.  ...  Using the newly derived generating function formulas, we are able to prove a bijective relationship between self-dual interval orders and upper-triangular matrices with no zero rows.  ...  Acknowledgment I am indebted to Michael Drmota for his insightful remarks and helpful suggestions.  ...

### Page 1165 of Mathematical Reviews Vol. 45, Issue 4 [page]

1973 Mathematical Reviews
This formula can be understood as follows: for each point x of [ choose a suitable set of conjugate variables (or a suitable quantifica- tion) and take for asymptotic expansion the one given by the usual  ...  The family of pure states which are asymptotically indistinguishable from the vacuum at large distances can be classified into subfamilies of super- selection sectors.  ...

### Page 589 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews
Let Sn,r be the set of all permutations of degree n with cycle lengths from a fixed set R and a,,z be the number of elements in Sy,x.  ...  This identity was used in an earlier paper of the author (“Covering a finite set by subsets”, Discrete Math., to appear] in counting collections of subsets of a set of indistinguishable elements whose  ...

### Probabilities of Sentences about Very Sparse Random Graphs

James F. Lynch
1992 Random structures & algorithms (Print)
W e consider random graphs with edge probability Pn-Q, where n is the number of vertices of the graph, / 3 > 0 is fixed, and a = 1 or a = (I + 1)/1 for some fixed positive integer I .  ...  We prove that for every firstorder sentence, the probability that the sentence is true for the random graph has an asymptotic limit.  ...  Clearly Ab does not depend on the choice of 3. For n 2 pr 2 (G) where G!=u(al ,..., a.) IGI=n xi), n) = C . . a l , . . . , ai are any distinct elements of the vertex set (0, 1, . . . , n -1).  ...

### Foundations of fuzzy sets: A nonstandard approach

C.A. Drossos
1990 Fuzzy sets and systems (Print)
It is argued in this paper that the common element of all new set-like concepts introduced is variability and vagueness.  ...  of something which is beyond the grasp of Cantorian set theory.  ...  If a = {an}, n e N, is an internal sequence of real numbers such that an ~-0 for all standard n, then there is a nonstandard v ~ N such that an ~ 0 for all n <~ v.  ...

### Asymptotic Conditional Probabilities: The Unary Case

Adam J. Grove, Joseph Y. Halpern, Daphne Koller
1996 SIAM journal on computing (Print)
(See Compton's overview 8] for an introduction to this work.) Assume, without loss of generality, that the domain is f1; : : :; Ng for some natural number N.  ...  The randomstructures method attempts to capture this intuition by considering a situation to be a structure|an isomorphism class of worlds.  ...  Finally, we would like to thank Andrew Granville for help with some of the details of Theorem 4.8.  ...

### The causes of bloat, the limits of health

Nick Mitchell, Gary Sevitsky
2007 Proceedings of the 22nd annual ACM SIGPLAN conference on Object oriented programming systems and applications - OOPSLA '07
We demonstrate the asymptotic nature of memory health: certain designs are limited in the health they can achieve, no matter how much the data size scales up.  ...  In some cases, large memory footprint helps accomplish an important functional, performance, or engineering requirement. A large cache, for example, may ameliorate a pernicious performance problem.  ...  Acknowledgments We would like to thank Tim Klinger, Palani Kumanan, and Edith Schonberg for their help with this work.  ...

### The causes of bloat, the limits of health

Nick Mitchell, Gary Sevitsky
2007 SIGPLAN notices
We demonstrate the asymptotic nature of memory health: certain designs are limited in the health they can achieve, no matter how much the data size scales up.  ...  In some cases, large memory footprint helps accomplish an important functional, performance, or engineering requirement. A large cache, for example, may ameliorate a pernicious performance problem.  ...  Acknowledgments We would like to thank Tim Klinger, Palani Kumanan, and Edith Schonberg for their help with this work.  ...

### Untestable Properties Expressible with Four First-Order Quantifiers [chapter]

Charles Jordan, Thomas Zeugmann
2010 Lecture Notes in Computer Science
We focus on the classification of first-order sentences based on their quantifier prefixes and vocabulary into testable and untestable classes.  ...  In particular, it is shown that there is an untestable graph property expressible with each of the following quantifier patterns: ∀∃∀∃, ∀∃∀ 2 , ∀ 2 ∃∀ and ∀ 3 ∃.  ...  Acknowledgments We would like to thank an anonymous referee for significantly improving Theorem 2 by eliminating one variable from each prefix, adding ∀∃∀∃, and also for simplifying the example following  ...
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