J. Arney and E. D. Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math., 103:269--294, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equivalent to the following process: draw a random variable N according to the geometric distribution of parameter 2x; if the value N = n is obtained, draw uniformly at random any of the possible words of size n. For the labelled case, consider the class K of all cyclic permutations, K = f[1] [1 2]; 1 2 3] 1; 3; 2] g. There are Kn = n 1) cyclic permutations of size n over the canonical set of labels f1; ng. The EGF is (3) b K(z) n 1) 1 1 z : The probability of drawing a cyclic permutation of some xed size n is then, 4) 1 log(1 x) a ....

....to the following process: draw a random variable N according to the geometric distribution of parameter 2x; if the value N = n is obtained, draw uniformly at random any of the possible words of size n. For the labelled case, consider the class K of all cyclic permutations, K = f[1] 1 2] [1 2 3]; 1; 3; 2] g. There are Kn = n 1) cyclic permutations of size n over the canonical set of labels f1; ng. The EGF is (3) b K(z) n 1) 1 1 z : The probability of drawing a cyclic permutation of some xed size n is then, 4) 1 log(1 x) a quantity de ....

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Arney, J., and Bender, E. D. Random mappings with constraints on coalescence and number of origins. Paci c Journal of Mathematics 103 (1982), 269-294.

....structures arise in a diversity of applications. For instance, statistics originally motivated the consideration of hierarchies where binary trees are also of some interest [23] functional graphs of various sorts intervene in cryptography as well as in some integer factorization methods, see [1, 9] for a treatment of their probabilistic properties. Generating functions. We next turn to the enumeration of decomposable structures via generating functions. If C is a class, we let Cn denote the number of objects in C having size n, and introduce the exponential generating function (egf) C(z) ....

....then a simple matter to find the singular expansion of GammaE near z = e GammaE 8 1 1 1 Gamma ez : The statement follows. 2 A similar result holds for mappings satisfying degree constraints (like specification K) whose probabilistic properties have been explored by Arney and Bender [1]. Set partitions and iterative structures. Set partitions correspond to the specification F = set(U ) where U = set(Z; card 1) denotes the class of blocks in partitions. We assume that U is given: to generate the shape of a block of size k in a partition, just output an undifferentiated set of ....

Arney, J., and Bender, E. D. Random mappings with constraints on coalescence and number of origins. Pacific Journal of Mathematics 103 (1982), 269--294.

....this works in a very similar way as in [12] There is also another way to get a more rigorous proof via the random mapping approach: When we consider random mapping built of planted plane trees instead of Cayley trees. Since this can be viewed as a special case of constrained random mappings (see [3, 4, 11]) it is easy to see that Theorem 2.1 still holds (with a different scaling parameter of course: p 2 instead of 2) Thus the explicit formulas [12, eq. 31 and eq. 32] can be used instead of the asymptotic ones below and the error estimates are much easier. But on the other hand, dealing with ....

ARNEY, J. and BENDER, E. A. (1982) Random mappings with constraints on coalescence and number of origins. Pacific J. Math. 103, 269--294.

....Then an element of Fn is called a random mapping. Each random mapping f can be represented by a functional graph, i.e. the graph consisting of the nodes 1; 2; n, and of the edges (i; f(i) i = 1; n. Various characteristics of random mappings have been studied. See e.g. [1, 3, 4, 6, 8, 9, 10, 12]. Arney and Bender [1] examined a more general model: They considered mappings such that the number of preimages of every point lies in a given set D of nonnegative integers (with 0 2 D) or, equivalently, the degrees of the nodes of the functional graph have to be in D. Let F D n denote the set ....

....random mapping. Each random mapping f can be represented by a functional graph, i.e. the graph consisting of the nodes 1; 2; n, and of the edges (i; f(i) i = 1; n. Various characteristics of random mappings have been studied. See e.g. 1, 3, 4, 6, 8, 9, 10, 12] Arney and Bender [1] examined a more general model: They considered mappings such that the number of preimages of every point lies in a given set D of nonnegative integers (with 0 2 D) or, equivalently, the degrees of the nodes of the functional graph have to be in D. Let F D n denote the set of those mappings on ....

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J. Arney and E. A. Bender, Random mappings with constraints on coalescence and number of origins, Pacific J. Math., 103 (1982), 269-294.

....There is a large probabilistic literature on the stucture of random mappings for uniform p. See e.g. 56, 3, 41, 65, 7] and papers cited there. The case when all of the p s but one are equal is studied in [83, 64, 18] Random p mappings for general p are studied in [23, 79, 48, 66, 6] See also [23, 31, 44, 45, 16, 17, 19, 50, 13, 75, 40] regarding various other models for random mappings. For each subset B of S S , the probability P (M 2 B) X m2B Y s2S pms (1) is the usual enumerator polynomial of B in variables p s ; s 2 S, as discussed in [27, p. 72] but with the constraints p s 0 and P s p s = 1. A formula for ....

J. Arney and E.A. Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math., 103:269--294, 1982. 38

....to know the exact value. We just have to evaluate these numbers asymptotically. We shalll show that this can be done by a singularity analysis of a proper bivariate generating function. It should be noted that some of our limit distributions on random mappings are well known (compare with [4, 16]) But our main goal is to provide a general method to derive such limit theorems. In particular, we use bivariate generating functions and singuarity analysis. Especially we are able to characterize the (up to now unknown) limit distribution of the number of those points with a xed number of ....

....(x 1 e ) k ; where c 00 = 0 and c 10 = 1 to see that A d (x; u t(x; u) has a representation of the kind A d (x; u t(x; u) 1 = g(x; u) h(x; u) p 1 ex; where g( 1 e ; 1) c 00 = 0 and h( 1 e ; 1) c 10 b( 1 e ) p 2. Hence we can apply Theorem 1. 2 14 Application 2 ([4]) Let r 0 be a xed integer and let Xn denote the number of points with j 1 (f g)j = r in mappings 2 Fn . Then Xn n p 2 n d N (0; 1) 46) where = 1 er and 2 = 1 (r 1) 2 ) 2 . Proof. Let F (p; x; u) xe p (u 1)x p r r : Then another application ....

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J. Arney and E.A. Bender, Random Mappings with constraints on coalescence and number of origins, Pac. J. Math. 103 (1982), 269-294.

....a proper limiting distribution or not. We are interested in the number of points N r with exactly r predecessors. For 2 Fn and x 2 f1; ng the elements V (x) k 0 k (fxg) 1) are called predecessors. Thus, for 2 Fn , N r;n ( jfx 2 f1; ng : jV (x)j = rgj: 2) In [11, 1] it was shown that EN r;n np r ; 1 The third author was partially supported by Contract No. 432 94 with the Bulgarian Ministry of Sciences. 1 where p r = r r 1 r e r = r 3 2 p 2 (1 O(r 1 ) 3) but the limiting distribution was left open. For constant r 1 Drmota and Soria ....

....and obtain d TV (N r;n ; Po(np r ) O( r 1 2 ) O 1 nr 3 2 = O(n 1 2 r 1 (1 r=n) 1 4 ) where = n 1 2 r 1 2 . Finally, if n=2 r n then we trivially have d TV (N r;n ; Po(np r ) O(np r ) O(n 1 2 r 1 ) 2 It should be mentioned, too, that Arney and Bender [1] discussed a slightly more general case. Let D be a set of non negative integers containing 0 and let F n denote those mappings 2 Fn such that number of immediate predecessors of a point j 1 (fxg)j is not arbitrary but must be contained in D. Hence the exponential generating function for ....

J. Arney and E.A. Bender, Random Mappings with constraints on coalescence and number of origins, Pacif. J. Math. 103 (1982), 269-294.

....a randomly chosen deranged mapping of [n 1] omits exactly k points from its image, 0 # k # n. In Section 5 we give an algorithm for computing these probabilities. The literature on random mappings is vast, see for example [6] although the fixed point free property does not appear often. In [2] the asymptotic distribution for many statistics on mappings is derived, including the normality of the image size. Of course, the number of elements omitted from the range di#ers from the latter only by a constant. It turns out we do not need information on the distribution of this statistic so ....

Jim Arney and Edward A. Bender, Random mappings with constraints on coalescence and number of origin, Pacific. J. Math. 103 (1982), 269--294.

....this works in a very similar way as in [12] There is also another way to get a more rigorous proof via the random mapping approach: When we consider random mapping built of planted plane trees instead of Cayley trees. Since this can be viewed as a special case of constrained random mappings (see [3, 4, 11]) it is easy to see that Theorem 2.1 still holds (with a di erent scaling parameter of course: p 2 instead of 2) Thus the explicit formulas [12, eq. 31 and eq. 32] can be used instead of the asymptotic ones below and the error estimates are much easier. But on the other hand, dealing with ....

ARNEY, J. and BENDER, E. A. (1982) Random mappings with constraints on coalescence and number of origins. Pacic J. Math. 103, 269-294.

....for the rho method for factoring, and works with the conjecture that the expected number of iterations is given as const= p V . For the case that the set of possible values for the in degree includes zero and at least one integer greater than one, this conjecture was proved by Arney and Bender [AB82]. Here, we have V = 1 in the case of (3.1) and V = 2=3 in the case of (3.2) so that the in degree method predicts that (3.1) requires p 2=3 = 0:82 times as many iterations as (3.2) 8 EDLYN TESKE 1 10 5 3 4 9 8 6 2 7 2 6 3 7 9 10 8 4 1 5 7 8 2 6 10 1 9 4 3 5 F ( Y ) ....

J. Arney and E. A. Bender, Random mappings with constraints on coalescence and number of origins, Pacific Journal of Mathematics 103 (1982), 269--294. MR 84h:05110

....necessary to know the exact value. We just have to evaluate these numbers asymptotically. We shall show that this can be done by a singularity analysis of a proper bivariate generating function. It should be noted that some of our limit distributions on random mappings are well known (compare with [4, 16]) But our main goal is to provide a general method to derive such limit theorems. In particular, we use bivariate generating functions and singularity analysis. Especially we are able to characterize the (up to now unknown) limit distribution of the number of those points with a fixed number of ....

....10 = 1 to see that A d (x, u t(x, u) has a representation of the kind A d (x, u t(x, u) 1 = g(x, u) h(x,u) # 1 ex, where g( 1 e , 1) c 00 = 0 and h( 1 e , 1) c 10 b( 1 e ) # 2. Hence, we can apply Theorem 1. 258 MICHAEL DRMOTA AND MICH ELE SORIA Application 2 (see [4]) Let r # 0 be a fixed integer, and let X n denote the number of points # with # 1 ( # ) r in mappings # #F n . Then X n n # # 2 n d #N(0, 1) 46) where = 1 er and # 2 = 1 (r 1) 2 ) 2 . Proof. Let F (p, x, u) xe p (u 1)x p r r . Then another ....

[Article contains additional citation context not shown here]

J. Arney and E. A. Bender, Random mappings with constraints on coalescence and number of origins, Pacific J. Math., 103 (1982), pp. 269--294.

....the right sides of (49) and (50) is easily verified directly. See [51, 31] for further results about p mappings. See [35, 2, 41] for results and further references to the literature for uniform p. The case when all of the p s but one are equal has also been studied in detail [55, 40, 8] See also [12, 21, 28, 29, 6, 7, 9, 33, 5] regarding various other models for random mappings. 5 Random Forests The proof of Theorem 1 is based on the following lemma: Lemma 18 For M a p mapping from S to S the distribution of the associated random forest F(M) is given by the formula P (F(M) f) jR(f)j 0 Y r2R(f) p r 1 A Y ....

J. Arney and E.A. Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math., 103:269--294, 1982.

....structures arise in a diversity of applications. For instance, statistics originally motivated the consideration of hierarchies where binary trees are also of some interest [23] functional graphs of various sorts intervene in cryptography as well as in some integer factorization methods, see [1, 9] for a treatment of their probabilistic properties. Generating functions. We next turn to the enumeration of decomposable structures via generating functions. If C is a class, we let Cn denote the number of objects in C having size n, and introduce the exponential generating function (egf) C(z) ....

....singular expansion of GammaE near z = e Gamma1 , GammaE p 2 8 1 (1 Gamma ez) 3=2 log 1 1 Gamma ez : The statement follows. 2 A similar result holds for mappings satisfying degree constraints (like specification K) whose probabilistic properties have been explored by Arney and Bender [1]. Set partitions and iterative structures. Set partitions correspond to the specification F = set(U ) where U = set(Z; card 1) denotes the class of blocks in partitions. We assume that U is given: to generate the shape of a block of size k in a partition, just output an undifferentiated set of ....

Arney, J., and Bender, E. D. Random mappings with constraints on coalescence and number of origins. Pacific Journal of Mathematics 103 (1982), 269--294.

....Knuth s book [23] for background information on random number generators. Random mappings are the subject of a vast collection of works; Mutafciev s survey [26] cites 113 references For general presentations, we direct the reader to the classic paper of Harris [19] the papers by Arney and Bender [1], and Stepanov [44] In this area, the contribution of the Russian school which uses essentially probabilistic methods, as shown by Kolchin s book Random Mappings [24] is notable. For completeness, we mention several recent papers not referenced in [24] namely [4, 7, 11, 20, 21, 30, 32] In ....

....to have been found in the 1950 s by a variety of methods including exact enumerations, discrete probability or generating functions. The paper by Harris [19] provides a first extensive approach to problems discussed in this section. Further results are given by Stepanov [44] or Arney and Bender [1], and our presentation follows similar lines. 3.1 Direct Parameters Let [ be a parameter of functional graph (or equivalently, mapping) such as the number of connected components. We introduce the quantities n = X 2Fn [ and 4(z) X n0 n z n n ; 18) called respectively ....

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J. Arney and E. D. Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math., 103:269--294, 1982.

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J. Arney and E. D. Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math., 103:269--294, 1982.

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J. Arney and E. D. Bender, Random mappings with constraints on coalescence and number of origins, Pacific J. Mathematics, 103 (1982), pp. 269--294.

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