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

1975
*
Journal of combinatorial theory. Series A
*

Ulam

doi:10.1016/0097-3165(75)90095-3
fatcat:uvlpdxhdsrhrnduts3e5cytjfm
*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 ...##
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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)

1997
*
Proceedings of the Edinburgh Mathematical Society
*

A

doi:10.1017/s0013091500023610
fatcat:vwljtohdnzdqjcr5hme6qvqhay
*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 ...##
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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

2011
*
Discrete Mathematics & Theoretical Computer Science
*

, including their size, magnitude, and

doi:10.46298/dmtcs.2932
fatcat:nx6jm5zdy5bmbi2ayluh4ex56q
*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]

2019
*
arXiv
*
pre-print

That is, any polynomial

arXiv:1906.10611v2
fatcat:idpzl4hmpjbwvivenxfvw6422a
*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

2012
*
Journal of combinatorial theory. Series A
*

In this paper, we present a new method to derive

doi:10.1016/j.jcta.2011.11.010
fatcat:mctpy3r3erhcrfrkjtrmjolfvy
*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

1992
*
Random structures & algorithms (Print)
*

W e consider random graphs with edge probability Pn-Q, where

doi:10.1002/rsa.3240030105
fatcat:jctrrksmsffrzfpzwpd2ifqik4
*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

1990
*
Fuzzy sets and systems (Print)
*

It is argued in this paper that

doi:10.1016/0165-0114(90)90027-4
fatcat:ovohpj4fyzb3hibi3pguqpjype
*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

1996
*
SIAM journal on computing (Print)
*

(See Compton's overview 8]

doi:10.1137/s0097539793257034
fatcat:l2jjt36yxrfkxntwwmkli2ibcq
*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. ...##
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The causes of bloat, the limits of health

2007
*
Proceedings of the 22nd annual ACM SIGPLAN conference on Object oriented programming systems and applications - OOPSLA '07
*

We demonstrate

doi:10.1145/1297027.1297046
dblp:conf/oopsla/MitchellS07
fatcat:iweb2xorajbpxbms2utraewmeq
*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

2007
*
SIGPLAN notices
*

We demonstrate

doi:10.1145/1297105.1297046
fatcat:t33qeaojrjhd3dcafs2cetvn74
*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]

2010
*
Lecture Notes in Computer Science
*

We focus on

doi:10.1007/978-3-642-13089-2_28
fatcat:ivrg7llrtbf27p5z2ztjkwy7ba
*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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