Nijenhuis, A. & Wilf, H. S. (1975), Combinatorial Algorithms, Academic Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....possible combinations of the elements of C without rechecking combinations previously examined. To accomplish this an algorithm was used that generates combinations in ascending numerical order. The original algorithm was obtained from an internet site that contained code from Nijenhuis and Wilf [11], and was modi ed so that the combinations were checked rst in order by size and then in ascending numerical order. In other words, potential covers of size 1 were rst formed by checking the c j of C in the order j = 1; 2; jCj. Next, potential covers of size 2 were formed by combining ....

Albert Nijenhuis and Herbert S. Wilf. Combinatorial Algorithms. Academic Press, Inc., Boston, Massachusetts, 2nd edition, 1978.

....blocks (K) and the blocksizes (X j ) with the correct distribution. To complete the task, it suces to transport this structure on a random permutation of the integers between 1 and N , where N = X 1 XK . The process distinctly di ers from the classical algorithm of Nijenhuis and Wilf [23] that requires tables of large integers. It is loosely related to a continuous model devised by Vershik [31] that can be interpreted as generating random set partitions based on S(x) Q j =j , i.e. by ordered block lengths, as a potentially in nite sequence of Poisson variables of ....

Nijenhuis, A., and Wilf, H. S. Combinatorial Algorithms, second ed. Academic Press, 1978.

....tournament. A construction sequence is also called a linear extension of a partial order. We use the following graph generating model for a random DAGs: 1. Select one of the n permutations with probability 1=n . A short algorithm generating random permutations with probability 1=n is given in [12]. The adjacency matrix of a graph that corresponds to the total order consists of a lower triangular matrix containing 1s, a diagonal containing 0s, and an upper triangular matrix containing 0s. The lower triangular matrix has n(n Gamma 1) 2 1s, each one corresponding to an arc. The associated ....

Nijenhuis, A. & Wilf, H. S. (1978). Combinatorial Algorithms. Academic Press: New York.

....code is cyclic with period the number of different codewords P. Otherwise, we say the Gray code is acyclic. Constructions for Gray codes can be found in [1] 3] while Gray codes have found application in diverse areas including coding theory [4] 5] and in the design of combinatorial algorithms [6], 7] Another common use of Gray codes is in reducing quantization errors in various types of analog todigital conversion systems [1] 8] As a typical example, a length n, period P Gray code can be used to record the absolute angular positions of a rotating wheel by encoding (e.g. optically) ....

A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms. New York: Academic Press, 1975.

....framework ; this technique allows to recursively generate the random objects by interpreting the enumeration recurrence relations. This generation e mail : dutour labri.u bordeaux.fr e mail : fedou unice.fr Figure 1: Complete binary trees. process have been formalized by Nijenhuis and Wilf [16], then by Hickey and Cohen in the case of context free languages [13] and by Greene within the framework of the labelled formal languages [12] Recently, Flajolet, Zimmermann and Van Cutsem have given a systematic approach for this method concerning their specications of structures [11] The ....

A. Nijenhuis and H.S. Wilf. Combinatorial algorithms. Academic Press Inc., 1979.

....ed in terms of a basic collection of general purpose combinatorial constructions. These constructions are precisely the ones that surface recurrently in modern theories of combinatorial analysis [4, 28, 30, 60, 61] and in systematic approaches to random generation of combinatorial structures [29, 51]. As a consequence, one obtains with surprising ease Boltzmann samplers covering an extremely wide range of combinatorial types. In most of the combinatorial literature so far, xed size generation has been the standard paradigm for the random generation of combinatorial structures, and a vast ....

....on the subject. There, either speci c bijections are exploited or general combinatorial decompositions are put to use in order to generate objects at random based on counting possibilities the latter approach has come to be known as the recursive method originating with Nijenhuis and Wilf [51], then systematized and extended by Flajolet, Zimmermann, and Van Cutsem in [29] Date: Version of January 1,2003. Submitted to Combinatorics, Probability, and Computing. In contrast, the basic principle of Boltzmann sampling is to relax the constraint of generating objects of a strictly xed ....

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Nijenhuis, A., and Wilf, H. S. Combinatorial Algorithms, second ed. Academic Press, 1978.

....we describe the first polynomial time algorithm that uniformly samples structural isomers given an empirical formula. Our isomer sampling algorithm is based on our algorithm for uniformly sampling unlabelled connected multigraphs with a given degree sequence. Previous work: Nijenhuis and Wilf [12] showed how to uniformly sample unlabelled rooted trees with a specified number of vertices. This approach was extended by Wilf [18] who showed how to uniformly sample free (unrooted) trees. Their algorithms are based on an inductive definition (i.e. a generating function) for the trees. This ....

.... equations to define the sets S r (n) in terms of the constructors and we will then argue that the definition is a specification (that is, the equations can be ordered in such a way that each equation depends only on sets previously defined) The set of equations is adapted from Nijenhuis and Wilf [12]. First, note that S 0 (0) fG 1 g and S 0 (n) for n 6= 0. Furthermore, S r (0) for r 6= 0. For n 0, we have S 1 (n) 0r Delta 1 i 0 Delta Delta Deltai Delta =n 1 S r (n 1; i 0 ; i r ; i r 1 1; i r 2 ; i Delta ) and n Delta S r (n) 1sr 1sdn d ....

A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms (2nd edition), Academic Press, 1978.

....can be syntactically constrained. In this context, the aim of our work is to generate more realistic random sequences by taking into account syntactic criteria as well as statistical ones. Our approach is based on the so named recursive method, which was initiated by Nijenhuis and Wilf [18] and then generalized and formalized by Flajolet, Zimmermann and Van Cutsem [9] Section 2 is devoted to a short presentation of this methodology within the framework of the algebraic languages. We present in section 3 a simple adaptation which allows to generate words in exact frequencies of the ....

A. Nijenhuis and H.S. Wilf. Combinatorial algorithms. Academic Press, New York, 2 edition, 1978.

....principle: Generate the root with the suitable probability distribution, then recursively generate the root subtrees. Several of the basic principles of this recursive top down approach have been formalized by Nijenhuis and Wilf in their reference book on combinatorial algorithms [28], by Hickey and Cohen in the case of context free languages [16] and under a fairly general setting by Greene within the framework of labelled grammars [14] The present work is in many ways a systematization and a continuation of the pioneering research of these authors. The class H of all ....

....time complexity O(n ) when applied to objects of size n; the boustrophedonic algorithms are based on a special search technique that proceeds in a bidirectional fashion and they exhibit O(n log n) worst case time complexity. The sequential method relies on existing technologies set forth by [14, 16, 28]; the boustrophedonic search extends to the realm of random generation an idea of Knuth for finding cycle leaders in permutations [19] Both methods appeal to precomputed numerical tables of size O(n) produced by a preprocessing phase of cost O(n to be effected once only. In the process of ....

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Nijenhuis, A., and Wilf, H. S. Combinatorial Algorithms. Academic Press, 1975.

....1;n code. We prove that every tree T of diameter 4 has a T code, and that no tree T of diameter 3 has a T code. Mathematical Reviews Subject Number: 05C45. 1 Introduction The utility of the ubiquitous binary reflected Gray code is undisputed. See, for example, the books of Nijenhuis and Wilf [5], Reingold, Nievergelt, and Deo [6] and Wilf [8] For certain applications, however, other Gray codes are desired. Many other Gray codes have been proposed, both for specific values of n and general constructions. For example, Goddyn, Lawrence, and Nemeth [3] motivated by an issue in the design ....

A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, Second Edition, Academic Press, 1978.

.... objects, has also done pioneering research on algorithms for generating combinatorial objects at random (that is, generating an element of a finite combinatorial set so that each element has the same probability of being generated as any other) see [CW] DW] GNW1] GNW2] NW1] [NW2], NW3] NW4] W1] W2] and [W3] for fruits of this research. This survey article describes a recent advance in the area of random generation, with applications to plane partitions, domino tilings, alternating sign matrices, and many other sorts of combinatorial objects. The algorithm is of ....

A. Nijenhuis and H. Wilf, Combinatorial Algorithms, Academic Press, 1975.

....more efficient algorithm is demonstrated to be random in a formal sense. A fourth algorithm that appears to use more randomness is then proved to be biased. The final section shows that a minor modification of this algorithm may be used to produce a random subset. The book by Nijenhuis and Wilf [4] is a wonderful catalogue of algorithms for solving combinatorial problems especially those involving sequences and subsets. For further study of the design and analysis of algorithms we also suggest [1] and [2] 2. Mixing the Values in X We begin by devising a method to imitate shuffling cards ....

....subsets. Finally we note that usually the ordering of the elements of a k subset of an n set is irrelevant but in some applications we might want them in increasing order. This can be done without first selecting the k elements and then sorting them, by algorithms which appear in [2] 3] and [4]. ....

A. Nijenhuis, H.S. Wilf, Combinatorial Algorithms, Academic Press, New York, 1978.

....a vast literature exists on the subject. There, either speci c bijections are exploited or general combinatorial decompositions are put to use in order to generate objects at random based on possibility counts this has come to be known as the recursive method originating with Nijenhuis and Wilf [12] and formalized by Flajolet, Zimmermann, and Van Cutsem in [8] In contrast, the basic principle of Boltzmann sampling is to relax the constraint of generating objects of a strictly xed size, and prefer to draw objects with a randomly varying size. As we shall see, normally, one can tune the ....

....Given the large number of combinatorial decompositions that have been gathered over the past two decades (see, e.g. 2, 7, 9] we estimate to perhaps a hundred the number of classical combinatorial structures that are amenable to ecient Boltzmann sampling. In contrast with the recursive method [3, 8, 12], memory requirements are kept to a minimum since only a table of constants of size O(1) is required. In forthcoming works starting with [5] we propose to demonstrate the versatility of Boltzmann sampling including: the generation of unlabelled multisets and powersets, the encapsulation of ....

Nijenhuis, A., and Wilf, H. S. Combinatorial Algorithms, second ed. Academic Press, 1978.

....dependence graph containing jobs 1; 2; and 3 in the first chain, and jobs 4 and 5 in separate chains. Without loss of generality we assume that two partitions are identical if the sets of component sizes for each partition are identical. The algorithm to select a random partition is described in [23]. This approach computes p(n; k) the number of partitions of n whose largest component is exactly k, and p(n) the number of partitions of n. Then the largest component is selected to be of size k with probability p(n; k) p(n) and this process is repeated with successively smaller components. ....

A. Nijenhuis and H. Wilf. Combinatorial Algorithms. Academic Press, 1975.

.... (T 1 ; T 2 ; c) where T 1 = T 1 1 ; T i 1 ) is the list anchored on v of a tree in T v(i) G 1 ; T 2 = T 1 2 ; T k Gammai 2 ) is the list anchored on v of a tree in T v(k Gammai) G 2 ; and c = ff 1 ; ff k Gammai 1 ) is a composition of i in k Gamma i 1 [NW78] To obtain the tree T 2 T v(k) G corresponding to such a triplet, merge the lists for T 1 and T 2 as specified by c each ff j indicates how many subtrees of the list of T 1 go between subtrees T j Gamma1 2 and T j 2 in the merged list. For example, Figure 17 shows how to find the T of ....

A. Nijenhuis and H. S. Wilf. Combinatorial algorithms. Academic Press, New York, 2nd edition, 1978.

.... (T 1 ; T 2 ; c) where T 1 = T 1 1 ; T i 1 ) is the list anchored on v of a tree in T v(i) G 1 ; T 2 = T 1 2 ; T k Gammai 2 ) is the list anchored on v of a tree in T v(k Gammai) G2 ; and c = ff 1 ; ff k Gammai 1 ) is a composition of i in k Gamma i 1 [NW78]. To obtain the tree T 2 T v(k) G corresponding to such a triplet, merge the lists for T 1 and T 2 as specified by c each ff j indicates how many subtrees of the list of T 1 go between subtrees T j Gamma1 2 and T j 2 in the merged list. For example, Figure 4 shows how to find the T of a ....

A. Nijenhuis and H. S. Wilf. Combinatorial algorithms. Academic Press, New York, 2nd edition, 1978.

....from L 2 ; now use the next ff 1 elements of L 1 , then one from L 2 . The last ff l 2 elements of L 1 follow the last element of L 2 in L. We then say that L is the result of merging L 1 ; L 2 using ff. Since the decomposition of n in k can be solved in n k Gamma 1 k Gamma 1 ways [NW78], there are M(l 1 ; l 2 ) l 1 l 2 l 2 acceptable results of the merge of lists L 1 ; L 2 , each identified with a specific decomposition. Observe that M(l 1 ; l 2 ) M(l 1 ; l 2 Gamma 1) M(l 1 Gamma 1; l 2 ) A table of size N Theta N can be constructed in O(N 2 ) time so that ....

A. Nijenhuis and H. S. Wilf. Combinatorial algorithms. Academic Press, New York, 2nd edition, 1978.

.... of six registers involved in computations and message passing in each of k systolic processors in order to program one combination generator [4, 5] Integer compositions can be generated in O(1) time per object using algorithm COMBGEN providing that known transformation from (n,k) combinations [34, 35, 37] is included in OUTPUT procedure. The solution presented for the combinations in the binary representation can be seen as a special case of parallel generation of n compositions of the integer k, with a restiction on component size (the component size is less or equal 1) Solving a more general ....

Nijenhius A., Wilf H.S.: Combinatorial Algorithms, Academic Press, New York 1978.

....January 2, 2001 Abstract Let be the 2 cycle (1 2) and oe the n cycle (1 2 Delta Delta Delta n) These two cycles generate the symmetric group S n . Let G n denote the directed Cayley graph Cay(f; oeg : S n ) Based on erroneous computer calculations, Nijenhuis and Wilf ( NiWi75] pg. 238 and [NiWi78], pg. 288) give as an exercise to show that G 5 does not have a Hamilton path. To the contrary, we show that G 5 is Hamiltonian. Furthermore, we show that G 6 has a Hamilton path. Our results illustrate how a little theory and some good luck can save a lot of time in backtracking searches. ....

....in part because of the existence of examples of directed Cayley graphs that are not Hamiltonian and there are even examples that don t have Hamilton paths. Among directed Cayley graphs without Hamilton paths, the sigma tau graph for n = 5 is often cited (e.g. Nijenhuis and Wilf [NiWi75] [NiWi78], Holszty nski and Strube [HoSt] Wilf [Wilf] Witte [Witt] and Witte and Gallian [WiGa] To be precise, let = 1 2) and oe = 1 2 Delta Delta Delta n) and let G n denote the Cayley graph Cay(foe; g : S n ) We will show that the sigma tau examples should not be used because G 4 has a ....

A. Nijenhuis and H.S. Wilf, Combinatorial Algorithms, 2nd ed., Academic Press, 1978.

....B.C. V8W 3P6, Canada January 2, 2001 Abstract Let be the 2 cycle (1 2) and oe the n cycle (1 2 Delta Delta Delta n) These two cycles generate the symmetric group S n . Let G n denote the directed Cayley graph Cay(f; oeg : S n ) Based on erroneous computer calculations, Nijenhuis and Wilf ([NiWi75], pg. 238 and [NiWi78] pg. 288) give as an exercise to show that G 5 does not have a Hamilton path. To the contrary, we show that G 5 is Hamiltonian. Furthermore, we show that G 6 has a Hamilton path. Our results illustrate how a little theory and some good luck can save a lot of time in ....

....is messier, in part because of the existence of examples of directed Cayley graphs that are not Hamiltonian and there are even examples that don t have Hamilton paths. Among directed Cayley graphs without Hamilton paths, the sigma tau graph for n = 5 is often cited (e.g. Nijenhuis and Wilf [NiWi75], NiWi78] Holszty nski and Strube [HoSt] Wilf [Wilf] Witte [Witt] and Witte and Gallian [WiGa] To be precise, let = 1 2) and oe = 1 2 Delta Delta Delta n) and let G n denote the Cayley graph Cay(foe; g : S n ) We will show that the sigma tau examples should not be used because G 4 ....

A. Nijenhuis and H.S. Wilf, Combinatorial Algorithms, 1st ed., Academic Press, 1975.

....designs in D. Thus we arrive at the simple algorithm given in Figure 6.1. This algorithm is obviously non polynomial, since the outer loop 1 9 will be executed v times. It is possible to enumerate all permutations so that successive permutations differ by a single transposition, see for example [97, 103]. Such an enumeration yields a constant amortised time (CAT) algorithm, and thus the overhead to generate all the permutations of V is O(v ) For each iteration of the loop 1 9, statement 3 will be executed n times and statements 5 and 6 will be executed n 2 times each. If we restrict v to be ....

Albert Nijenhuis and Herbert S. Wilf. Combinatorial Algorithms. Academic Press, second edition, 1978.

....shall revisit this point brie y in Section 5.5. The reader is also referred to Jerrum s papers [25] and [27] One situation in which counting technology can be used for sampling orbits is when generating functions for enumerating orbits can be eciently evaluated. For example, Nijenhuis and Wilf [41] used Equation 3.1 (see Section 3.4) to obtain a polynomial time algorithm for sampling rooted unlabelled trees. Their algorithm is given in Figure 1 (see also [48] Note that this is an exactly uniform sampling algorithm its output distribution is exactly the uniform distribution on orbits. ....

A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, 2nd Edition, Academic Press, (1978). Leslie Ann Goldberg

....The problems in the class #P Complete can be thought of as decision problems (often from the complexity class NP Complete) with the additional requirement of determining not just the existence of solutions, but determining the exact number of solutions. For the full details, see [GJ79] We know [NW75] an algorithm for P0 1. Fact 3 A 0 1 permanent of size n Theta n can be computed with O(n 2 n ) arithmetic operations. 4 DNA Realizations of 0 1 Permanent Evaluation In the first subsection, we give a very simple algorithm for evaluating or approximating a 0 1 permanent. This algorithm ....

Albert Nijenhuis and Herbert S. Wilf. Combinatorial Algorithms. Academic Press, New York, 1975.

.... still an important open problem to find an efficient algorithm for generating a planar graph uniformly at random [HP73, DVW96, Sch97] Several techniques have been developed to build sequential algorithms for generating combinatorial structures according to some predefined probability distribution [NW78]. In Chapter 2 we look at the issues involved in finding parallel algorithms for sampling combinatorial objects uniformly at random. In some cases trivial parallelisation of a sequential algorithm solves the problem quite efficiently. In some others the nature of the problem seems to prevent ....

....requirement of getting exact counting information or an output which is exactly uniform is relaxed then both types of problems sometimes become solvable in polynomial time. Sequential algorithms exist for the uniform generation of several classes of combinatorial objects. The reader is referred to [NW78] for a large collection of early results and to surveys like [Tin90, Sin93, DVW96, Wil97] for a few more up to date results. In this chapter we address the problem of determining the parallel complexity of some uniform generation problems on graphs. 26 Very little seems to be known about this ....

A. Nijenhuis and H.S. Wilf. Combinatorial Algorithms. Academic Press, New York, 1978.

....several classes of languages including regular, context free (c.f. for short) and more generally languages accepted by one way nondeterministic auxiliary push down automata (1 NAuxPDA) Recall that the random generation is a classical problem widely studied in the literature of the last decades [22, 27, 19, 18, 14, 12]. In particular several (sequential) algorithms have been proposed for the random generation of strings in regular and context free languages [16, 24, 14, 11, 15] The problem is particularly interesting in the c.f. case because these languages can codify a wide variety of combinatorial ....

A. Nijenhuis and H .S. Wilf. Combinatorial Algorithms. Academic Press, 2nd edition, 1978.

....advantageous to have an ecient method to generate random binary trees with some xed number n of nodes in order to test the program. Random generation of binary trees is a special case of random generation of combinatorial structures. A seminal work on this area is the book of Nijenhuis and Wilf [9] (see also [13] The work of Nijenhuis and Wilf was later generalized by Flajolet et al. 4] A general method for various kinds of trees was introduced by Alonso et al. 1] see also [2] A recent survey on random generation of binary trees is [7] This note discusses an algorithm for ....

A. Nijenhuis and H.S. Wilf, Combinatorial Algorithms. Second Edition. Academic Press, 1978.

....: ng up to size bn=2c in order of increasing cardinality. We can view these subsets as ordered tuples with the indices in each tuple sorted in increasing order. Algorithm SEARCH enumerates all tuples of a given length in lexicographic order. A good source of such combinatorial algorithms is [14]. Naturally, the corresponding piece of code in a factorizer, which could actually be iterative, will be more complicated because when a true factor is found the indices of the modular factors comprising it should be removed from further consideration. 2.1 Memory Stacks The fact that test tuples ....

Nijenhuis, A., and Wilf, H. S. Combinatorial Algorithms, second ed. Academic Press, 1978. 16

.... of Decomposable Structures Using Floating Point Arithmetic Alain Denise , Paul Zimmermann y Th eme 2 G enie logiciel et calcul symbolique Projet Eur eca Rapport de recherche n3242 Septembre 1997 16 pages Abstract: The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8] is extended here to floating point arithmetic. The resulting ADZ method enables one to generate decomposable data structures both labelled or unlabelled uniformly at random, in expected O(n 1 ffl ) time and space, after a ....

.... 83 19 Antenne de Metz, technopole de Metz 2000, 4 rue Marconi, 55070 METZ Telephone : 33) 87 20 35 00 Telecopie : 33) 87 76 39 77 G en eration al eatoire uniforme de structures d ecomposables en arithm etique flottante R esum e : La m ethode r ecursive mise au point par Nijenhuis et Wilf [15] et syst ematis ee par Flajolet, Van Cutsem et Zimmermann [8] est ici etendue a l utilisation de nombres flottants. La m ethode qui en d ecoule, appel ee ADZ, permet de g en erer al eatoirement et uniform ement des structures d ecomposables etiquet ees ou non en temps et espace moyens ....

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Nijenhuis, A., and Wilf, H. S. Combinatorial Algorithms, second ed. Academic Press, 1978.

....closely related. Section 3 presents a new algorithm for generating red black trees. There are algorithms for generating many other kinds of data structures and discrete objects, such as binary trees (Lucas, van Baronaigien, and Ruskey 1993) B trees (Kelsen 1996) and partitions of a finite set (Nijenhuis and Wilf 1978). Using the methodology presented in this paper, these algorithms could be used to test applications in the area of expression trees, databases, and data structures for disjoint sets. Frankl and Doong (Doong and Frankl 1994) develop black box drivers and use algebraic specifications for output ....

Nijenhuis, A. and H. Wilf (1978). Combinatorial Algorithms (second ed.). Academic Press.

....even less the speed of this convergence. The aim of our paper is to ll this gap. Note that the results of Aldous, Drmota Gittenberger, and Tak acs are actually about general simple trees. Rooted labeled trees are a special case of simple trees, but an important one (see [28] p. 389 390, 33] [34] or [44] 1.2 Results It is part of history of probability that: Pr(m x) X 1 k 1 (1 4k 2 x 2 )e 2k 2 x 2 and, for r 1: r = E(m r ) 2 r=2 r(r 1) r 2 (r) 1 = r 2 ; see [12] 13] 27] 32] 7] 46] We shall say that m is theta distributed by ....

A. Nijenhuis, H.S. Wilf (1975) Combinatorial algorithms. Academic Press, New YorkLondon.

.... B x the subset of elements of B at the i j th place for lexicographic order (1 # j # t) Here is a trivial property which will be helpful for computing the compositions, since we know a constant worst case time complexity algorithm to derive the successor of a partition (see Nijenhuis and Wilf [8]) Proposition 2 For given n, p and #, every partition in # p #n###n# is equivalent to a unique composition in ## #n# p#. Proof. The proof is immediate if we see that, given a partition y in # p #n # ## n#, the corresponding composition x in # # #n# p# is such that x i is nothing but the number ....

Nijenhuis, A. and Wilf, H. S. (1975). Combinatorial Algorithms. Academic Press.

....generating combinatorial objects uniformly at random. The work lies in the recursive method framework; this method is to generate recursively random objects by endowing a recurrence formula with a probabilistic interpretation. This generation process has been first formalized by Nijenhuis and Wilf [11, 14, 15]. They have a general approach. They base the recursive procedure on an acyclic directed rooted graph with a terminal vertex and numbered edges, graph which depends on the family of objects. The recursive method has been also formalized by Hickey and Cohen in the special case of context free ....

Nijenhuis, A. and Wilf, H. S. (1975). Combinatorial Algorithms, 2nd ed. Academic Press.

....Stirling numbers, hooklengths, cycle index polynomials, and the partition function. There are also functions for generating all objects of a certain type, ranking and unranking them, and picking an object at random. Many of these constructions are taken from the book of Nijenhuis and Wilf [3] where they are presented as Fortran programs. 3 Some of the Combinatorica functions were written with an eye towards illustrating certain programming techniques, rather than making them as efficient as possible. For example, DistinctPermutations, which lists all the different permutations of a ....

....complexity of generating all 2 element subsets of an n set, I m liable to get blank stares. A better approach is to time the Combinatorica function KSubsets which lists all such combinations. For added effect, we can make a plot of these times for n = 10; 20; 100 by calling In[3]: ListPlot[ Table[n,Timing[KSubsets[Range[n] 2] 1,1] fn,10,100,10g] which yields the graph in Figure 1. FIGURE 1 ABOUT HERE 4 Now when I ask what function would approximate the data points, I get a response that easily motivates the equation n(n Gamma 1) 2 = O(n 2 ) III. Graph ....

Albert Nijenhuis and Herbert S. Wilf, Combinatorial Algorithms, 2nd ed., Academic Press, New York, NY, 1978. 7

.... of permanents we refer to Minc (1978) Several ways of computing permanents are compared in Chapter 7 of Minc (1978) The best method seems to be an algorithm based on inclusion exclusion due to Ryser (see Ryser (1963) His algorithm, together with some improvements, is also described in Nijenhuis and Wilf (1975). The Ryser algorithm was designed for matrices with scalar entries, but it also works, although less efficient, for our matrices P and e P with monomial entries. The computer algebra system Maple has a built in function permanent, which computes permanents using minor expansions. We experimented ....

Nijenhuis, A., and H.S. Wilf (1975). Combinatorial algorithms . London Academic Press.

....design problem is a difficult combinatorial optimization problem. There are P P i=0 ( Gamma1) i (P Gammai) N i (P Gammai) different variations of clustering N users into P clusters so that no cluster is empty [17] and P P Gamma2 different variations of inter cluster spanning trees [18]. Thus, the solution space size of the optimization problem is P N P=1 P P Gamma2 P P i=0 ( Gamma1) i (P Gammai) N i (P Gammai) The clustering problem itself is harder from the known graph partition problem which is classified in [9] as NPComplete. As mentioned in [7] a simplified ....

A. Nijenhuis, Combinatorial Algorithms. New York:academic, 1975.

....tournament. A construction sequence is a linear extension of a partial ordering. We use the following graph generating model for random DAGs: 1. Select one of the (n ) permutations with probability 1= n ) A short algorithm generating random permutations with probability 1= n ) is given in [34]. The adjacency matrix of a graph that corresponds to a total ordering can always be arranged such that the lower triangular matrix contains 1s, and that the diagonal and the upper triangular matrix contain 0s. The lower triangular matrix has n(n Gamma 1) 2 1s, each one corresponding to an arc. ....

Nijenhuis, A. & Wilf, H. S. (1978). Combinatorial Algorithms. New York: Academic Press.

....partition of an MDG is to enumerate all of its partitions and select the partition with the largest MQ value. This algorithm is not practical for MDGs with a large (over 15) number of modules, because the number of partitions of a graph grows exponentially 1 with respect to its number of nodes [14]. Thus, Bunch uses more efficient search algorithms to discover acceptable sub optimal results. These algorithms are based on hill climbing and genetic algorithms. Figure 2 shows the main window of the Bunch tool. This window prompts the user to specify a clustering algorithm (e.g. ....

A. Nijenhuis and H. S. Wilf. Combinatorial Algorithms. Academic Press, 2nd edition, 1978.

....the following is an plane partition of 10: 17 21 331 In other words as you go up a column, the parts are nonincreasing and as you go across a row the parts are nonincreasing. In this case the set of primes and the synthesis operation are not as obvious. The details can be found in chapter 12 of [13]. Lemma 3 In any prefab P, let d n be the number of prime objects of order n and let a n be the total number of objects of order n, then # X n=0 a n x n = # Y j=1 1 (1 x j ) d j . The proof of this is similar to the proof of the validity of Euler s generating function for ....

....primes. Consider the generating function for plane partitions: # Y j=1 1 (1 x j ) j . This generating function looks like one for a prefab with j primes of order j,foreachj # 0. It is indeed that, and the identification was made by Bender and Knuth. A description of their work is in [13]. A related object is a solid partition, in which the parts are nonincreasing along each of 3 dimensions. It has been shown that solid partitions are not prefabs and it is currently an open problem to enumerate the solid partitions of the integer n. Now let s use the previous lemma and consider ....

Albert Nijenhuis and Herbert S. Wilf, Combinatorial Algorithms, 2nd Ed., Academic Press, New York, 1978.

....GF (q) then the probability that f; g are relatively prime is 1 Gamma 1=q. We give explicit involutions for the pentagonal sieve theorem, generalizing earlier mappings found by Bressoud and Zeilberger. 1 The Main Theorem The natural context in which our results lie is that of prefabs. A prefab ([1, 3, 4]) P is a combinatorial structure in which each object is uniquely representable as a product ( synthesis ) of powers of prime objects, and in which there is an order function j j 2 Z which satisfies j 0 j = j j j 0 j. We denote the primes of P by p 1 ; p 2 ; Examples of ....

A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, 2nd ed., Academic Press, New York, 1978.

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Nijenhuis, A. & Wilf, H. S. (1975), Combinatorial Algorithms, Academic Press.

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A. Nijenhuis, H. S. Wilf. Combinatorial Algorithms, Academic Press, New York, 1978.

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A. Nijenhaus and H.S. Wilf, Combinatorial Algorithms, 2nd. ed., Academic Press, New York, (1978).

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A. Nijenhuis and H. Wilf, Combinatorial Algorithms. Academic Press, 2 ed., 1978.

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A. Nijenhaus and H.S. Wilf, Combinatorial Algorithms, 2nd. ed., Academic Press, New York, (1978).

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A. Nijenhuis and H. S. Wilf, (1978). Combinatorial Algorithms. 2nd Edition. Academic Press, New York.

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A. Nijenhaus adn H.S. Wilf, Combinatorial Algorithms, 2nd. ed., Academic Press, New York, (1978).

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A. Nijenhuis and H. S. Wilf. Combinatorial Algorithms. Academic Press, 2nd edition, 1978.

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A. Nijenhuis and H. S. Wilf. Combinatorial algorithms. Academic Press, New York, NY, 1975.

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A. Nijenhuis, and H. S. Wilf, Combinatorial Algorithms, 2nd ed., Academic Press, New York, 1978.

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