R. P. Stanley, Generating functions, in Studies in Combinatorics, M.A.A. Studies in Mathematics, vol. 17 (1978), ed. G.-C. Rota, The Mathematical Association of America, pp. 100141.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for detailed definitions. Though more in the style of computer science s data types, it is consistent with common practice in the combinatorial analysis of labelled structures; there the product is sometimes called the labelled product or the partitional product. The interested reader can consult [12, 17, 29, 33, 34, 37] for background information on these classical notions. Definition 1 Let T = T 0 ; T 1 ; Tm ) be an (m 1) tuple of classes of combinatorial structures. A specification of T is a collection of m 1 equations, with the ith equation being of the form T i = Psi i (T 0 ; T 1 ; Tm ) ....

....card = k) C(z) A (z) k C = cycle(A; card = k) C(z) A (z) k : ii) Given a specification, the corresponding enumerating sequences up to size n are all computable in O(n ) arithmetic operations. Proof. We refer to standard texts on combinatorial analysis, see for instance [12, 17, 29, 33, 34, 37]. The details of the O(n ) algorithms are given in [11, 14] and they also result from the standard specifications of the next section. 2 Observe that, by summation, one further derives translation rules like C = set(A; card k) C(z) j=0 j C = set(A; card k) C(z) e ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100-- 141.

....degree of the k recurrences involving all the a (j) n . Thus the use of several sequences a (j) n leads to much simpler and combinatorially more appealing relations. That generating functions can significantly simplify combinatorial problems is shown by the following example. It is taken from [349], and is a modification of a result of Klarner [229] and Polya [321] This example also shows a more complicated derivation of explicit generating functions than the simple ones presented so far. Example 6.5. Polyomino enumeration [349] Let a n be the number of n square polyominoes P that are ....

....problems is shown by the following example. It is taken from [349] and is a modification of a result of Klarner [229] and Polya [321] This example also shows a more complicated derivation of explicit generating functions than the simple ones presented so far. Example 6.5. Polyomino enumeration [349]. Let a n be the number of n square polyominoes P that are inequivalent under translation, but not necessarily under rotation or reflection, and such that each row of P is an unbroken line of squares. Then a 1 = 1, a 2 = 2, a 3 = 6. We define a 0 = 0. It is easily seen that a n = # (m 1 m 2 ....

[Article contains additional citation context not shown here]

R. P. Stanley, Generating Functions, in Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17., G--C. Rota, ed., Math. Ass. of America, 1978, pp. 100--141.

....( T(n 2)x n 2 ) cn 2 ( n 2) cn 2 c n 2 # # # = # j#T e T (j) x j j 1 # n#T x n n # n #T (s T (n) 1) x n n = e FT (x) e x # n #T s T (n) x n n and the proposition follows. The reader well acquainted with the exponential formula (see, e.g. [S2]) might wonder whether it is possible to use it to prove Proposition 2. Indeed, that is the case, but we have preferred to give a more direct proof to avoid unnecessary terminology. 4. Three lemmas. This section consists of three lemmas with proofs of a rather technical nature. The reader is ....

R. Stanley, Generating functions, in Studies in Combinatorics, G.-C. Rota, ed., MAA Studies in Mathematics, 1978, pp. 100--141.

....is admissible if it admits a translation into generating functions. The following two theorems are well known under one form or the other. They embody a powerful collection of combinatorial constructions. For detailed definitions, the reader is referred to modern treatments of the subject [15, 42, 43, 72, 74, 81] or to the paper [32] where a similar system of notations is developed. Theorem 1 (Admissible constructions for OGF s) For unlabelled structures, the constructions of union, cartesian product, sequence, cycle, set, multiset, substitution are admissible. The translations into ordinary generating ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

....for detailed definitions. Though more in the style of computer science s data types, it is consistent with common practice in the combinatorial analysis of labelled structures; there the product is sometimes called the labelled product or the partitional product. The interested reader can consult [12, 17, 29, 33, 34, 37] for background information on these classical notions. Definition 1 Let T = T 0 ; T 1 ; Tm ) be an (m 1) tuple of classes of combinatorial structures. A specification of T is a collection of m 1 equations, with the ith equation being of the form T i = Psi i (T 0 ; T 1 ; Tm ) ....

....card = k) C(z) A k (z) k C = cycle(A; card = k) C(z) A k (z) k : ii) Given a specification, the corresponding enumerating sequences up to size n are all computable in O(n 2 ) arithmetic operations. Proof. We refer to standard texts on combinatorial analysis, see for instance [12, 17, 29, 33, 34, 37]. The details of the O(n 2 ) algorithms are given in [11, 14] and they also result from the standard specifications of the next section. 2 Observe that, by summation, one further derives translation rules like ( C = set(A; card k) C(z) P k j=0 A j (z) j C = set(A; card k) ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100-- 141.

....expression. 4.2. Some Applications Let us take here the occasion of a few examples to discuss some further features of Ananas. The next three examples are all taken from combinatorial enumerations: E1) Trees of cycles of cycles of beads; E2) Involutive permutations; E3) Children s Rounds of [Stanley 1978]; E4) Bell numbers counting partitions of n. E1] equivalent(1 2 (1 sqrt(1 4 log(1 (1 log(1 (1 z) 1 2 1 2 exp(exp( 1 4) exp( 3 8) 3 2 1 2 1 (n ( exp(exp( 1 4) exp( 1) 1) n 1 2 ( exp(exp( 1 4) exp( 1) 1) Pi ) etc . 1 2 n) exp(exp( 1 4) 1) 1) ....

R. P. Stanley [1978]. "Generating Functions," in Studies in Combinatorics, edited by G-C. Rota, M. A. A. Monographs, 1978.

....section and the next one is an informal reminder of basic properties of analytic functions intended for asymptotic analysis. For a detailed treatment, we refer the reader to one of the many excellent treatises on the subject, like the books by Dieudonn e [2] Henrici [5] Knopp [6] Titchmarsh [15], or Whittaker and Watson [16] Analytic functions. A function f(z) of the complex variable z is analytic at a point z = a if it is defined in a neighbourhood of z = a and is representable there by a convergent power series expansion f(z) X n0 f n (z Gamma a) n : 4:4) The functions ....

....modulus of a function analytic at 0 are called dominant singularities. Dominant singularities play an essential role in the asymptotic analysis of coefficients of generating functions, as we shall see in the next section. 1 For a precise discussion, see [2, p. 229] 6, vol. 1, p. 82] or [15]. 4.3. SINGULARITIES 9 Rephrasing an earlier observation (a converging Taylor series is analytic) we get that a function analytic at 0 which is represented by a Taylor series at z = 0 with a finite radius of convergence always has (at least) one singularity on its circle of convergence. In ....

[Article contains additional citation context not shown here]

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

....X i=0 0 d i X j=0 p ij z j 1 A y (i) z) 0; 8) where we assume that the leading coefficients p id i are different from 0. Suppose the formal power series f(z) P n0 fn z n satisfies (8) Then by substituting f into (8) and equating coefficients of z n , it is well known (see [10]) that one gets a linear recurrence with polynomial coefficients: r X i=0 d i X j=0 p ij (n i Gamma j) Delta Delta Delta (n Gamma j 1)f n i Gammaj = 0; 9) valid for all n, with the convention that f k = 0 when k 0. We shall denote M = max(fd i Gamma i; i = 1; rg) then ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

....but not both. A unifying framework comprising both types was proposed by Joyal [49] in 1981. Finally, a systematic exposition of combinatorial enumerations in this context is the subject of a book by Jackson and Goulden [39] Other relevant references are Comtet s book [24] and Stanley s works [76, 78]. On the analytical side, the tradition of relating analytic properties of a function to asymptotic properties of its Taylor coefficients is older. Its roots lie in part in classical analysis (e.g. Darboux s method) and in part in analytic number theory (e.g. the additive theory of partitions) ....

....basic elements by means of a fixed collection of standard set theoretic constructions. To be more precise, the constructions operate in a parallel manner in two different universes, the unlabelled and the labelled universe a dichotomy that is familiar from classical combinatorial analysis [38, 39, 49, 76]. An issue to be discussed is the notion of well definedness of specifications; this is dealt with in Section 3.2. The situation there resembles that of context free languages with respect to properness of grammatical specifications. unlabelled labelled union [ union union [ union ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

....singularities are not algebraic, some transfer theorems are still available. These theorems [11, 6] enable us to use the same algorithm as before with a slight modification in step 3, when we translate the local expansion into the coefficients expansion. Example 13: Children rounds are defined [16] as sets of cycles with a child in the middle of each cycle. The number of children rounds with n children gives rise to a nice expansion (gamma is Euler s fl constant, fl 0:5772156) equivalent( 1 z) z) z,n,5) y We recall the classical notation [z n ]f(z) meaning the coefficient of z ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

.... geometry, and even the interpretation of perturbative expansions in statistical physics [7] The purpose of this paper is to re examine these problems in the light of recent general methods of analytic combinatorics [14, 28] First thanks to symbolic methods developed by various schools [4, 14, 15, 18, 28, 29, 32], there is a systematic and purely formal correspondence between combinatorial constructions and generating functions. In this way, specifications of combinatorial structures can be translated automatically into generating function equations. This approach is, as we propose to show, especially ....

Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

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R. Stanley, Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, 1978, pp. 100--141.

No context found.

R. Stanley, Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, Washington, DC, 1978, pp. 100--141.

No context found.

R. Stanley, Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, 1978, pp. 100--141.

No context found.

R. Stanley, Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, 1978, pp. 100--141.

No context found.

R. Stanley, Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, Washington, DC, 1978, pp. 100--141.

No context found.

R. P. Stanley, Generating functions, in Studies in Combinatorics, M.A.A. Studies in Mathematics, vol. 17 (1978), ed. G.-C. Rota, The Mathematical Association of America, pp. 100141.

No context found.

R. P. Stanley, Generating functions, in Studies in Combinatorics, M.A.A. Studies in Mathematics, vol. 17 (1978), ed. G.-C. Rota, The Mathematical Association of America, pp. 100141.

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