Abstract
We consider the multiset construction of decomposable structures with component generating function $C(z)$ of alg-log type, $\textit{i.e.}$, $C(z) = (1-z)^{-\alpha} (\log \frac{1}{ 1-z})^{\beta}$. We provide asymptotic results for the number of labeled objects of size $n$ in the case when $\alpha$ is positive and $\beta$ is positive and in the case $\alpha = 0$ and $\beta \geq 2$. The case $0<-\alpha <1$ and any $\beta$ and the case $\alpha > 0$ and $\beta = 0$ have been treated in previous papers. Our results extend previous work of Wright.
Highlights
Certain combinatorial structures can be decomposed into smaller and simpler objects called components
The exponents α and β are called the algebraic exponent and the logarithmic exponent, respectively. This type of component generating function belongs to the class of alg-log component generating functions, which was introduced by Flajolet and Soria [10] in their study of the number of components in combinatorial structures
For the exp-alglog class, Dong, Gao, Panario, and Richmond [5] considered the cases 0 < −α < 1 with any β, and α > 0 and β = 0. Their asymptotic analysis used two different approaches: the first case was done with singularity analysis, and the second was done using the saddle point method
Summary
Certain combinatorial structures can be decomposed into smaller and simpler objects called components. The exponents α and β are called the algebraic exponent and the logarithmic exponent, respectively This type of component generating function belongs to the class of alg-log component generating functions, which was introduced by Flajolet and Soria [10] in their study of the number of components in combinatorial structures. For the exp-alglog class, Dong, Gao, Panario, and Richmond [5] considered the cases 0 < −α < 1 with any β, and α > 0 and β = 0 Their asymptotic analysis used two different approaches: the first case was done with singularity analysis, and the second was done using the saddle point method.
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