Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis

Abstract

We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of "petals" in all the flowers (petals' weight) and the number of edges in the petals of all the flowers (flowers' weight). We give explicit expressions of their generating functions and deduce general formulas for the asymptotic growth of their coefficients and the expectations of their concentrated distributions.

Figures (1)

• Figure 1: Three examples of trees with flowers, one for each class $\mathcal{R}$, $\mathcal{S}$ and $\mathcal{T}$ seen in this work, respectively. For all of them the tree's weight is $20$. (a) A tree with flowers on the leaves (there are 13 leaves), its size is $409$, the flowers' weight is $389$ and the petals weight is 89. (b) A tree with flowers everywhere but on the root. (c) A tree with flowers everywhere.

Theorems & Definitions (15)

• Proposition 1: Generating functions of trees with flowers
• proof
• Proposition 2
• Proposition 3
• Proposition 4: Cumulative generating functions of parameters on the flowers of trees
• Theorem 5: Single singularity analysis of trees with flowers
• proof
• Theorem 6: Asymptotics for meromorphic functions of $1$-trees with flowers
• proof
• Theorem 7: Asymptotics of $1$-trees with non-plane flowers on the leaves and bounded petal size