Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

Pemantle, Robin; Wilson, Mark C.

doi:10.1137/050643866

Cited by 87 publications

(120 citation statements)

References 51 publications

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“…Theorem 1.3 is proved in Section 3 using a result of R. Pemantle and M.C. Wilson [18,19,20] concerning multivariate asymptotics.…”

Section: Historical Remarkmentioning

confidence: 99%

“…The following result of Pemantle and Wilson appeared as Theorem 3.1 in [18], as Corollary 3.21 in [19], and as Theorem 9.5.7 in [20].…”

Section: Definition 1 a Point (X Y) ∈ V Is Called Strictly Minimal mentioning

confidence: 99%

“…To that end, we note a result of Clark and Ehrenborg (see We will apply a result of Pemantle and Wilson [18,19,20] to the coefficients A r,s when (r, s) ∈ S ε0 (cf. equation (1.4)), which will provide asymptotics on these coefficients.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…Theorem 3.1 (Pemantle, Wilson [18,19,20]). Let F = G/H : C 2 → C be a meromorphic function with variety of singularities V = V F .…”

Section: Definition 1 a Point (X Y) ∈ V Is Called Strictly Minimal mentioning

confidence: 99%

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Asymptotics of the extremal excedance set statistic

Andrade

Lundberg

Nagle

2015

European Journal of Combinatorics

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Abstract. For non-negative integers r and s with r + s = n − 1, let [b r a s ] denote the number of permutations π ∈ Sn which have the property that π(i) > i if, and only if, i ∈ [r] = {1, . . . , r}. Answering a question of Clark and Ehrenborg (2010), we determine asymptotics on [b r a s ] when r = (n − 1)/2 :We also determine asymptotics on [b r a s ] for a suitably related r = Θ(s) → ∞. Our proof depends on multivariate asymptotic methods of R. Pemantle and M. C. Wilson.We also consider two applications of our main result. One, we determine asymptotics on the number of permutations π ∈ Sn which simultaneously avoid the vincular patterns 21-34 and 34-21, i.e., for which π is order-isomorphic to neither (2,1,3,4) nor (3, 4, 2, 1) on any coordinates 1 ≤ i < i+1 < j < j +1 ≤ n. We also determine asymptotics on the number of n-cycles π ∈ Cn which avoid stretching pairs, i.e., those for which 1 ≤ π(i) < i < j < π(j) ≤ n.

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“…Theorem 1.3 is proved in Section 3 using a result of R. Pemantle and M.C. Wilson [18,19,20] concerning multivariate asymptotics.…”

Section: Historical Remarkmentioning

confidence: 99%

“…The following result of Pemantle and Wilson appeared as Theorem 3.1 in [18], as Corollary 3.21 in [19], and as Theorem 9.5.7 in [20].…”

Section: Definition 1 a Point (X Y) ∈ V Is Called Strictly Minimal mentioning

confidence: 99%

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…Theorem 3.1 (Pemantle, Wilson [18,19,20]). Let F = G/H : C 2 → C be a meromorphic function with variety of singularities V = V F .…”

Section: Definition 1 a Point (X Y) ∈ V Is Called Strictly Minimal mentioning

confidence: 99%

See 2 more Smart Citations

Asymptotics of the extremal excedance set statistic

Andrade

Lundberg

Nagle

2015

European Journal of Combinatorics

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“…More recently central limit behavior has been shown to follow when the generating function is rational and obeys a smoothness hypothesis [PW04,PW08], or in certain cases when the generating function is algebraic [Gre15].…”

Section: Introductionmentioning

confidence: 99%

Multivariate CLT follows from strong Rayleigh property

Ghosh¹,

Liggett²,

Pemantle³

2017

2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Let (X1, . . . , X d ) be a random nonnegative integer vector. Many conditions are known to imply a central limit theorem for a sequence of such random vectors, for example, independence and convergence of the normalized covariances, or various combinatorial conditions allowing the application of Stein's method, couplings, etc. Here, we prove a central limit theorem directly from hypotheses on the probability generating function f (z1, . . . , z d ). In particular, we show that the f being real stable (meaning no zeros with all coordinates in the open upper half plane) is enough to imply a CLT under a nondegeneracy condition on the variance. Known classes of distributions with real stable generating polynomials include spanning tree measures, conditioned Bernoullis and counts for determinantal point processes. Soshnikov [Sos02] showed that occupation counts of disjoint sets by a determinantal point process satisfy a multivariate CLT. Our results extend Soshnikov's to the class of real stable laws. The class of real stable laws is much larger than the class of determinantal laws, being defined by inequalities rather than identities. Along the way we investigate the related problem of stable multiplication.

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Step‐Growth Polymerized Systems of Type “A1+A2+A3”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

Hillegers¹,

Slot

2021

Macro Theory & Simulations

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might break up later on in the reaction process, and the two now disconnected members of the pair might find other partners. The initial number (molar amounts) of monomers of type A1, A2, and A3 are denoted by n [1], n[2], and n[3], respectively. Then, n tot = n[1] + n[2] + n [3] is the total number of monomeric units, andis the total number (molar amounts) of A groups available to form chemical bonds. The fraction of f tot is denoted by p (named degree of conversion, or extent of reaction) that has formed a bond. It is assumed that after a while, at each value of p, 0 ≤ p ≤ 1, the reaction process has reached a dynamic equilibrium in which the number of bonds does not change over time, and the total number of polymeric molecules remains at the stable value of n tot -1 2 p f tot . When a satisfactory level is reached, the reaction process is frozen by lowering the temperature and/or by removing the condensation by-product. In modeling the reaction process and the formation of polymeric molecules, we make the usual assumptions: i) all reactions occur with equal probability, independent of the size of the molecules involved and independent of the status of other A groups on the same monomeric unit; ii) cycles do not form, i.e., no reaction happens between A groups on the same molecule; and iii) there are no other reactions than between A groups. [1,2] The system "A1+A2+A3," that we study in this paper, is the most simple representative of the more general step-growth polymerizing systems of type "Afi," whereby the functionality (number of A groups) on each monomer species is not limited to three. It is our intention, in future studies, to extend the calculation method developed here for "A1+A2+A3" to step-growth polymerizing systems of type "AfiBgi." Those systems are characterized by monomers bearing A groups and/ or B groups; an A group reacts only with a B group, and vice versa.These step-growth polymerizing systems, "Afi" and "AfiBgi," and the many variations thereof are widely studied in the chemical literature and are of great interest to the polymer industry. Examples of commercially relevant systems "Afi" with AA condensation and water as condensation by-product, are: linear polymers, "A2": POM (polyoxymethylene of paraformaldehyde)Step-growth polymerizing systems of type "A1+A2+A3" are considered. The monomers bear one (A1), two (A2), or three (A3) identical reactive sites. In the reactor vessel, at a given degree of conversion, a wide range of polymeric molecules has formed, differing in both molecular size and in branching structure, determined by the laws of probability. In a slice of the molecular size distribution, all polymeric molecules have the same size (i.e., are built up by the same number of monomeric units), but differ in number and position of branching points (A3's). A method is presented to calculate the path length distribution for each such slice. Here, path length is the number of chemical bonds in the path connecting two monomeric units in the molecule. The shape and moments of this di...

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Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions

Cited by 87 publications

References 51 publications

Asymptotics of the extremal excedance set statistic

Asymptotics of the extremal excedance set statistic

Multivariate CLT follows from strong Rayleigh property

Step‐Growth Polymerized Systems of Type “A1+A2+A3”: A Method to Calculate the Bivariate (Molecular Size) × (Path Length) Distribution

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