Central and local limit theorems applied to asymptotic enumeration. IV: Multivariate generating functions. (English) Zbl 0755.05004
Summary: P. Flajolet and M. Soria [J. Comb. Theory, Ser. A 53, No. 2, 165-182 (1990; Zbl 0691.60035)] discussed some general combinatorial structures in which central limit theorem and exponential tail results hold. In this paper, we shall use P. Flajolet and A. Odlyzko’s “transfer theorems” [SIAM J. Discrete Math. 3, No. 2, 216- 240 (1990; Zbl 0712.05004)] to extend E. A. Bender and L. B. Richmond’s [J. Comb. Theory, Ser. A 34, 255-265 (1983; Zbl 0511.05003)] central and local limit theorems to a wider class of generating functions which will cover the above-mentioned combinatorial structures. The local limit theorem provides more accurate asymptotic information and implies the superexponential tail results.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
Keywords:
generating function; limit theorem; asymptotic enumeration; exponential tail; combinatorial structuresReferences:
| [1] | Bender, E. A.; Richmond, L. B., Central and local limit theorems applied to asymptotic enumeration II: multivariate generating functions, J. Combin. Theory Ser. B, 34, 255-265 (1983) · Zbl 0511.05003 |
| [2] | Flajolet, P.; Odlyzko, A., Singularity analysis of generating functions, SIAM J. Discrete Math., 3, 216-240 (1990) · Zbl 0712.05004 |
| [3] | Flajolet, P.; Soria, M., Gaussian limiting distributions for the number of components in combinatorial structures, J. Combin. Theory Ser. A, 53, 165-182 (1990) · Zbl 0691.60035 |
| [4] | Flajolet, P.; Soria, M., General combinatorial schemas with Gaussian limit distributions and exponential tails (1989), preprint |
| [5] | Renyi, A., Probability Theory (1970), North-Holland: North-Holland Amsterdam · Zbl 0206.18002 |
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