The fine- and generative spectra of varieties of monounary algebras

Abstract

In this paper we present recursive formulas to compute the fine spectrum and generative spectrum of each of the varieties of monounary algebras. Hence, an asymptotic or log-asymptotic estimation for the number of n-generated and n-element algebras is given in every variety of monounary algebras. These results provide infinitely many examples of spectra with different orders of magnitude that are asymptotically bigger than any polynomial and smaller than any exponential function.

1 Introduction

For a variety of algebras $\mathcal V$ let $g_{\mathcal V}(n)$ denote the number of n-generated algebras in $\mathcal V$, and let $f_{\mathcal V}(n)$ denote the number of n-element algebras in $\mathcal V$ up to isomorphism. The sequences $(g_{\mathcal V}(n))_{n\in \mathbb {N}}$ and $(f_{\mathcal V}(n))_{n\in \mathbb {N}}$ are called the generative spectrum and the fine spectrum of ${\mathcal V}$, respectively. For a detailed introduction into generative- and fine spectra, see [1]. The asymptotic behaviour of these sequences for certain varieties of algebras is often strongly related to the algebraic properties of the structures in the variety. For example, a finitely generated variety $\mathcal {V}$ of groups is nilpotent if and only if $g_{\mathcal {V}}(n)$ is at most polynomial, and a finite ring R generates a variety with at most exponential generative spectrum if and only if the square of the Jacobson radical of R is trivial [1]. The infinite counterpart of our problems is widely investigated in model theory. The famous Vaught conjecture says that the cardinality of the set of non-isomorphic models of any first-order theory in a countable language is either countable or continuum. In [4, 5] the conjecture is verified for varieties of algebras.

A monounary algebra, $\mathcal {A}=(A;u)$ is an algebra with a single unary operation u. The theory of monounary algebras is well-developed, for a recent monograph see [9]. All varieties of monounary algebras were classified by Jacobs and Schwabauer [7]. Every variety of monounary algebras can be defined by a single identity. The variety $\mathcal {V}_{k,d}$ is defined by the equation $u^{k+d}(x)=u^k(x)$, and the variety $\mathcal {V}_{k}$ is defined by the equation $u^k(x)=u^k(y)$, where $u^0={{\,\mathrm{id}\,}}, u^1=u$, and in general $u^{n+1}=u\circ u^n$.

In [6] a formula was obtained for the number of n-element monounary algebras. Let $M_n$ and $C_n$ denote the number of monounary algebras and connected monounary algebras, respectively. It was shown in [6] that $\log _\alpha C_n\sim \log _\alpha M_n\sim n$ for a constant $\alpha \approx 2.95576$. In our terminology, this result shows the log-asymptotic behaviour of the fine spectrum of the variety $\mathcal {V}_{0,0}$, the class of all monounary algebras. In [1], several results were proven about the growth rate of the generative spectrum of varieties. In many cases, the spectrum is at most polynomial (e.g., pure sets, vector spaces over finite fields) or at least exponential (e.g., Boolean algebras, semilattices). The variety $\mathcal {V}_2$ is mentioned in [1] as an interesting example for a locally finite variety whose generative spectrum is bigger than any polynomial and smaller than any exponential function. It was explicitly calculated there that the number of non-isomorphic n-generated algebras in $\mathcal {V}_2$ is bigger than p(n) and smaller than $(n+1)^2p(n)$, where p(n) is the number of partitions of n. An asymptotic formula for the fine spectrum of $\mathcal {V}_{2}$ and the log-asymptotic behaviour of the fine spectrum of $\mathcal {V}_{k}$ were determined in [10] for all k.

The goal of the present paper is to obtain a recursive formula for the generative spectrum and fine spectrum of all the varieties $\mathcal {V}_{k,d}$ and $\mathcal {V}_{k}$, and to determine the log-asymptotic behaviour of these sequences. In some cases, we can even determine the asymptotic behaviour or provide an explicit formula for the fine- and generative spectra. The main results are presented in Theorems 5.1, 5.2, 6.1, 6.3.

2 Description of the varieties

2.1 Monounary algebras as directed graphs

The function u defines a directed graph on A. Let $G_A=(A;E)$, the vertex set is A and the edges are $E=\{(a,u(a))\,\mid \, a\in A \}$. In $G_A$ every vertex has out-degree 1, and every directed graph G with all vertices having out-degree 1 defines a monounary algebra on its vertex set, where u(a) is the single vertex such that (a, u(a)) is an edge in G. Hence, a monounary algebra can be identified with a directed graph, where each vertex has out-degree 1. This identification gives rise to a number of notions. The algebra (A; u) is connected if the graph $G_A$ is connected. More generally, the connected components of (A; u) are the connected components of $G_A$.

Let B be a connected component of (A; u) with a cycle of length d. Then the graph $G_B$ contains exactly one cycle. If we remove edges of this cycle in $G_B$, then we obtain d rooted trees whose edges are directed towards the root. These trees give a partition of the component B into d sets. We call this partition the cyclic partition of B. The roots of the trees in the cyclic partition are the vertices of the cycle, and an element a in the connected component is in the rooted tree with root r if and only if r is the first element of the cycle in the sequence $(u^k(a))_{k=0}^{\infty }$.

2.2 Varieties of monounary algebras

The notion of an equational class goes back to Birkhoff [2], who has shown that a class of algebras can be defined by a set of equations if and only if the class is closed under taking homomorphic images, subalgebras and (possibly infinite) direct products. Such classes are also called varieties. According to [7], every variety of monounary algebras can be defined by a single equation. Based on this result, they have given the following exhaustive list of varieties of monounary algebras.

• The varieties $\mathcal {V}_{k,d}$ are defined by the equation $u^{k}(x)=u^{k+d}(x)$, for $k\ge 0, d\ge 1$. An algebra (A; u) is in $\mathcal {V}_{k,d}$ if and only if every connected component B of (A; u) contains a cycle whose length divides d, and every rooted tree in the cyclic partition of B is of depth at most k. In particular, every n-generated algebra in $\mathcal {V}_{k,d}$ has at most $n(k+d)$ elements, showing that the generative spectra of these varieties are indeed sequences of integers.

• The class of all monounary algebras is $\mathcal {V}_{0,0}$ defined by the equation $x=x$. As there are infinitely many n-generated algebras in $\mathcal {V}_{0,0}$ for all n, the generative spectrum of this variety is not defined. In view of [6], the fine spectrum of $\mathcal {V}_{0,0}$ is $(M_n)_{n\in \mathbb {N}}$ and $\log M_n \sim n\log \alpha$, where $\alpha \approx 2.95576$.

• The varieties $\mathcal {V}_{k}$ are defined by the equation $u^k(x)=u^k(y)$, for $k\ge 1$. The classes $\mathcal {V}_{k}$ consist of connected monounary algebras. If $(A;u)\in \mathcal {V}_{k}$, then the cycle of (A; u) is a loop, i.e., a single vertex r with $u(r)=r$. Thus $G_A$ is a rooted tree with root r. This leads to the following combinatorial description: $(A;u)\in \mathcal {V}_{k}$ if and only if $G_A$ is a rooted tree of depth at most k. In particular, the number of n-element algebras in $\mathcal {V}_{k}$ equals to the number of n-element rooted trees of depth at most k. The log-asymptotic behaviour of the fine spectrum of $\mathcal {V}_k$ was determined in [10]. The log-asymptotic behaviour of the generative spectrum can be computed in a similar fashion. As a preliminary observation, we note that the two spectra are not very far from each other, because an n-generated algebra has at most $nk+1$ elements in this variety, and every n-element algebra is n-generated. The detailed computation and the results are presented in Sections 5 and 6.

• $\mathcal {V}_{0}$ consists of the isomorphism type of the one-element algebra, and it is defined by the equation $x=y$. The problem of computing the generative spectrum and fine spectrum of $\mathcal {V}_{0}$ is trivial.

Note that pseudovarieties of monounary algebras are described in [8].

3 Generating functions

Definition 3.1

Throughout the paper $\log$ denotes the natural logarithm function, and $L_m$ denotes the m-fold iterated logarithm function, namely $L_m(x)=\log \log \dots \log x$. The exponential function $e^x$ is denoted by $\exp (x)$. The number of positive divisors of n is denoted by $\tau (n)$.

We use the symbols $\sim , o(.), O(.)$ in the standard way. Given two series $a,b: \mathbb {N}\rightarrow \mathbb {R}$ with $b_n\ne 0$ for all but finitely many n, we put $a_n\sim b_n$ if $\lim _{n\rightarrow \infty } \frac{a_n}{b_n} = 1$, $a_n=o(b_n)$ if $\lim _{n\rightarrow \infty } \frac{a_n}{b_n} = 0$, and $a_n=O(b_n)$ if $\frac{a_n}{b_n}$ is bounded. Furthermore, the expression “$\log$-asymptotic behaviour of $a_n$” refers to an asymptotic estimation of the sequence $(\log a_n)_{n\in \mathbb {N}}$, i.e., a sequence $(b_n)_{n\in \mathbb {N}}$ with $\log a_n \sim b_n$. Note that finding such a sequence $(b_n)_{n\in \mathbb {N}}$ is usually less demanding than finding a sequence $(c_n)_{n\in \mathbb {N}}$ with $a_n\sim c_n$, or even one with $a_n=O(c_n)$: if $a_n=O(c_n)$, $c_n> 0$ for all but finitely many n and $\log a_n\rightarrow \infty$, then clearly $\log a_n \sim b_n$ for $b_n=\log c_n$.

Definition 3.2

• For $k\ge 0$, $f_k(n)$ is the number of non-isomorphic n-element algebras in $\mathcal {V}_k$, which equals to the number of n-element rooted trees of depth at most k. The generating function of the sequence $(f_k(n))_{n=1}^{\infty }$ is denoted by $F_k(x)=\sum _{n=1}^{\infty } f_k(n)x^n$.

• For $k\ge 0$, $g_k^*(n)$ is the number of rooted trees of depth at most k with exactly n leaves, i.e., vertices with in-degree 0 in this context. Note that, by definition, the singleton tree has one leaf. The generating function of the sequence $(g_k^*(n))_{n=1}^{\infty }$ is denoted by $G_k^*(x)=\sum _{n=1}^{\infty } g_k^*(n)x^n$.

• For $k\ge 0$, $g_k(n)$ is the number of rooted trees of depth at most k with at most n leaves, which equals to the number of n-generated algebras in $\mathcal {V}_k$. The generating function of the sequence $(g_k(n))_{n=1}^{\infty }$ is denoted by $G_k(x)=\sum _{n=1}^{\infty } g_k(n)x^n$.

• For $k\ge 0, d\ge 0$, $f_{k,d,{{\,\mathrm{con}\,}}}(n)$ is the number of non-isomorphic connected n-element algebras in $\mathcal {V}_{k,d}$, which equals to the number of n-element digraphs with a directed cycle of length dividing d, such that by omitting the edges of the cycle the graph is partitioned into rooted trees of depth at most k, and the edges of each tree are directed towards the root. The generating function of the sequence $(f_{k,d,{{\,\mathrm{con}\,}}}(n))_{n=1}^{\infty }$ is denoted by $F_{k,d,{{\,\mathrm{con}\,}}}(x)=\sum _{n=1}^{\infty } f_{k,d,{{\,\mathrm{con}\,}}}(n)x^n$.

• For $k\ge 0, d\ge 0$, $f_{k,d}(n)$ is the number of non-isomorphic n-element algebras in $\mathcal {V}_{k,d}$. The generating function of the sequence $(f_{k,d}(n))_{n=1}^{\infty }$ is denoted by $F_{k,d}(x)=\sum _{n=1}^{\infty } f_{k,d}(n)x^n$.

• For $k\ge 0, d\ge 0$, $g^*_{k,d,{{\,\mathrm{con}\,}}}(n)$ is the number of non-isomorphic connected n-generated but not $(n-1)$-generated algebras in $\mathcal {V}_{k,d}$, which equals to the number of digraphs with exactly n leaves, containing a directed cycle of length dividing d, such that by omitting the edges of the cycle the graph is partitioned into rooted trees of depth at most k, and the edges of each tree are directed towards the root. The generating function of the sequence $(g^*_{k,d,{{\,\mathrm{con}\,}}}(n))_{n=1}^{\infty }$ is denoted by $G^*_{k,d,{{\,\mathrm{con}\,}}}(x)=\sum _{n=1}^{\infty } g^*_{k,d,{{\,\mathrm{con}\,}}}(n)x^n$.

• For $k\ge 0, d\ge 0$, $g^*_{k,d}(n)$ is the number of non-isomorphic n-generated but not $(n-1)$-generated algebras in $\mathcal {V}_{k,d}$. The generating function of the sequence $(g^*_{k,d}(n))_{n=1}^{\infty }$ is denoted by $G^*_{k,d}(x)=\sum _{n=1}^{\infty } g^*_{k,d,{{\,\mathrm{con}\,}}}(n)x^n$.

• For $k\ge 0, d\ge 0$, $g_{k,d}(n)$ is the number of non-isomorphic n-generated algebras in $\mathcal {V}_{k,d}$. The generating function of the sequence $(g_{k,d}(n))_{n=1}^{\infty }$ is denoted by $G_{k,d}(x)=\sum _{n=1}^{\infty } g_{k,d,{{\,\mathrm{con}\,}}}(n)x^n$.

There are several recurrence formulas for the sequences defined in Definition 3.2, which we use to obtain the asymptotic estimations. All of these formulas can be written up in terms of the power series of the sequences.

Lemma 3.3

The formal power series defined in Definition 3.2 satisfy the following formulas coefficient-wise for all integers $k,d\ge 0$.

• (1) $F_{k+1}(x)=x\exp (\sum _{m=1}^{\infty } \frac{1}{m}F_k(x^m))$.

• (2) $G^*_{k+1}(x)=\exp (\sum _{m=1}^{\infty } \frac{1}{m}G^*_k(x^m))+x-1$.

• (3) $F_{k,1,{{\,\mathrm{con}\,}}}(x)=F_{k}(x)$.

• (4) $\frac{1}{d}(F_{k,1,{{\,\mathrm{con}\,}}}(x))^d\le F_{k,d,{{\,\mathrm{con}\,}}}(x)\le \sum _{t|d} (F_{k,1,{{\,\mathrm{con}\,}}}(x))^t$.

• (5) $F_{k,d}(x)=\exp (\sum _{m=1}^{\infty } \frac{1}{m}F_{k,d,{{\,\mathrm{con}\,}}}(x^m))-1$.

• (6) $G^*_{k,1,{{\,\mathrm{con}\,}}}(x)=G^*_{k}(x)$.

• (7) $\frac{1}{d}(G^*_{k,1,{{\,\mathrm{con}\,}}}(x))^d\le G^*_{k,d,{{\,\mathrm{con}\,}}}(x)\le \sum _{t|d}(G^*_{k,1,{{\,\mathrm{con}\,}}}(x))^t$.

• (8) $G^*_{k,d}(x)=\exp (\sum _{m=1}^{\infty } \frac{1}{m}G^*_{k,d,{{\,\mathrm{con}\,}}}(x^m))-1$.

Proof

Item (1) is shown in [10], see Theorem 2.2. The proof of item (2) is analogous.

Items (3) and (6) are straightforward from Definitions 3.2.

The proofs of items (5) and (8) are based on a similar argument, thus we only show item (5). For $1\le i\le n$ let $\mu _i$ be the number of i-element connected components in the algebra (A; u). Up to isomorphism, (A; u) is determined by the isomorphism types of its connected components. There are $\left( {\begin{array}{c}f_{k,d,{{\,\mathrm{con}\,}}}(j)+\mu _j-1\\ \mu _j\end{array}}\right)$ ways to choose $\mu _j$ connected algebras in $\mathcal {V}_{k,d}$ of size j. Thus $f_{k,d}(n)=\sum _{\sum i\mu _i=n} \prod _{j=1}^{n}\left( {\begin{array}{c}f_{k,d,{{\,\mathrm{con}\,}}}(j)+\mu _j-1\\ \mu _j\end{array}}\right)$. According to the generalised binomial theorem, for every $|x|<1$ we have that $(1-x^j)^{-f_{k,d,{{\,\mathrm{con}\,}}}(j)}= \sum _{\mu _j=0}^{\infty }\left( {\begin{array}{c}-f_{k,d,{{\,\mathrm{con}\,}}}(j)\\ \mu _j\end{array}}\right) \cdot \left( -x^j\right) ^{\mu _j} =\sum _{\mu _j=0}^{\infty }\left( {\begin{array}{c}f_{k,d,{{\,\mathrm{con}\,}}}(j)+\mu _j-1\\ \mu _j\end{array}}\right) x^{j\mu _j}$. Thus for $n\ge 1$, $f_{k,d}(n)$ equals to the n-th coefficient in the power series $\prod _{j=1}^{\infty } (1-x^j)^{-f_{k,d,{{\,\mathrm{con}\,}}}(j)}$. The constant term of the power series $\prod _{j=1}^{\infty } (1-x^j)^{-f_{k,d,{{\,\mathrm{con}\,}}}(j)}$ is 1. Hence, $F_{k,d}(x)=\prod _{j=1}^{\infty } (1-x^j)^{-f_{k,d,{{\,\mathrm{con}\,}}}(j)}-1= \exp (\sum _{j=1}^{\infty }\log (1-x^j)^{-f_{k,d,{{\,\mathrm{con}\,}}}(j)})-1= \exp (\sum _{j=1}^{\infty } f_{k,d,{{\,\mathrm{con}\,}}}(j) (-\log (1-x^j)))-1$. By expanding $-\log (1-x)$ we obtain $F_{k,d}(x)= \exp (\sum _{j=1}^{\infty } f_{k,d,{{\,\mathrm{con}\,}}}(j) \sum _{m=1}^{\infty } \frac{1}{m}x^{jm})-1= \exp (\sum _{m=1}^{\infty } \frac{1}{m}F_{k,d,{{\,\mathrm{con}\,}}}(x^m))-1$.

Finally, the proofs of items (4) and (7) are similar, thus we only show item (4). Let (A; u) be a connected algebra in $\mathcal {V}_{k,d}$ such that the length of its cycle is t. Then t|d. Let $r_1, \ldots , r_t$ be an enumeration of the elements of the cycle of (A; u) such that $u(r_1)=r_2, \ldots , u(r_t)=r_1$. This enumeration depends on the choice of $r_1$. By omitting the edges of the cycle of (A; u), we obtain a partition of $G_A$ into t rooted trees of depth at most k. The isomorphism type of the rooted tree with root $r_i$ is denoted by $x_i$. Let us assign the t-tuple $(x_1, \ldots , x_t)$ to (A; u). Depending on the choice of $r_1$, it might be possible to assign more than one tuple to (A; u). As there are t ways to choose $r_1$ with t|d, the number of tuples assigned to an algebra in $\mathcal {V}_{k,d}$ is at most d. Up to isomorphism, the algebra (A; u) is uniquely determined by any of its assigned tuples. For t|d let $S_{k,t}(n)$ be the set of tuples $(x_1, \ldots , x_t)$ of isomorphism types of rooted trees with n elements altogether and of depth at most k. Let $s_{k,t}(n)=|S_{k,t}(n)|$. Every tuple in $S_{k,t}(n)$ is assigned to an n-element algebra in $\mathcal {V}_{k,d}$. Hence, the above argument shows that $\frac{1}{d} s_{k,d}(n)\le f_{k,d,{{\,\mathrm{con}\,}}}(n)\le \sum _{t|d} s_{k,t}(n)$. The number of tuples $(x_1, \ldots , x_t)\in S_{k,t}(n)$ such that a rooted tree with isomorphism type $x_i$ has $\mu _i$ vertices is $\prod _{i=1}^{t} f_{k,1,{{\,\mathrm{con}\,}}}(\mu _i)$. Thus $s_{k,t}(n)= \sum _{\mu _1+\cdots +\mu _t=n}\prod _{i=1}^{t} f_{k,1,{{\,\mathrm{con}\,}}}(\mu _i)$, which is the n-th coefficient in the power series $(F_{k,1,{{\,\mathrm{con}\,}}}(x))^t$. $\square$

The techniques used in Lemma 3.3 can be found in [3]. The following theorem is from [10]. Although in [10] these assertions were only shown for specific values of the parameters, the proof works in full generality without any modification.

Theorem 3.4

Let $(a_n)_{n\in {\mathbb {N}}}$, $(b_n)_{n\in {\mathbb {N}}}$ be sequences of positive integers, and let $A(x)=\sum _{n=1}^{\infty } a_nx^n$ and $B(x)=\sum _{n=1}^{\infty } b_nx^n$ be the generating functions of these sequences. Assume that $B(x)=\exp (\sum _{m=1}^{\infty } \frac{1}{m}A(x^m))$.

• (1) If $\log a_n\sim C\sqrt{n}$ for some $C>0$, then $\log b_n\sim \frac{C^2}{4}\frac{n}{\log n}$.

• (2) For $k\ge 1$, if $\log a_n\sim C\frac{n}{L_{k}(n)}$ for some $C>0$, then $\log b_n\sim C\frac{n}{L_{k+1}(n)}$.

We note that in [10] B(x) is of the form $x\cdot \exp (\sum _{m=1}^{\infty } \frac{1}{m}A(x^m))$, that item (1) is proved for $a_n=f_2(n)$, $C=\pi \sqrt{\frac{2}{3}}$, and that item (2) is proved for $a_n=f_{k-2}(n)$, $k>4$, $C=\frac{\pi ^2}{6}$.

4 Auxiliary computations

Lemma 4.1

Let $K,C\in \mathbb {R}^+$, $s\in \mathbb {R}$. Let $a_n\sim Kn^s\exp (C\sqrt{n})$, and let $b_n=\sum _{i=1}^{n}a_i$. Then $b_n\sim \frac{2K}{C}n^{s+1/2}\exp (C\sqrt{n})$.

Proof

As $a_n\rightarrow \infty$ and $a_n\sim Kn^s\exp (C\sqrt{n})$, we have that $b_n=\sum _{i=1}^{n}a_i \sim \sum _{i=1}^{n} Ki^s\exp (C\sqrt{i})$. Let $n_0=\lfloor n-2n^{2/3}+n^{1/3}\rfloor$. Then for all $i\le n_0$ we have $C\sqrt{i}\le C\sqrt{n}-Cn^{\frac{1}{6}}$ and $Ki^s\le Kn^s + K$. Thus $b_{n_0} = O(\sum _{i\le n_0} Ki^s\exp (C\sqrt{i}))\le O((Kn^s + K)\exp (C\sqrt{n})\exp (-Cn^{1/6}))=o(a_n)$. Hence, by the monotonicity of $n^s$, and by using $n^s\sim n_0^s$, we obtain that $b_n\sim \sum _{n_0<i\le n}Ki^s\exp (C\sqrt{i})\sim n^{s+1/2}\sum _{n_0<i\le n}\frac{K}{\sqrt{i}}\exp (C\sqrt{i})\sim n^{s+1/2}\sum _{i=1}^{n}\frac{K}{\sqrt{i}}\exp (C\sqrt{i})$.

The function $\frac{K}{\sqrt{x}}\exp (C\sqrt{x})$ is monotone for x large enough, thus replacing the sum $\sum _{i=1}^{n}\frac{K}{\sqrt{i}}\exp (C\sqrt{i})$ with the integral $\int \limits _{x=1}^{n}\frac{K}{\sqrt{x}}\exp (C\sqrt{x})$ introduces an error of order of magnitude $O(\frac{\exp (C\sqrt{n})}{\sqrt{n}})$. As $\frac{2K}{C}\exp (C\sqrt{x})$ is a primitive function of $\frac{K}{\sqrt{x}}\exp (C\sqrt{x})$, we have $\sum _{i=1}^{n}\frac{K}{\sqrt{i}}\exp (C\sqrt{i})= O(\frac{\exp (C\sqrt{n})}{\sqrt{n}})+ \int \limits _{x=1}^{n}\frac{K}{\sqrt{x}}\exp (C\sqrt{x})= O(\frac{\exp (C\sqrt{n})}{\sqrt{n}}) + \frac{2K}{C}\exp (C\sqrt{n})$. Consequently, the error $O(\frac{\exp (C\sqrt{n})}{\sqrt{n}})$ is negligible, and we obtain $\sum _{i=1}^{n}\frac{K}{\sqrt{i}}\exp (C\sqrt{i}) \sim \frac{2K}{C}\exp (C\sqrt{n})$, and $b_n\sim n^{s+1/2}\sum _{i=1}^{n}\frac{K}{\sqrt{i}}\exp (C\sqrt{i}) \sim \frac{2K}{C}n^{s+1/2}\exp (C\sqrt{n})$. $\square$

Lemma 4.2

Let $d\in \mathbb {N}$. Then $\max _{n_1+\cdots +n_d=n}\sum _{i=1}^{d} \sqrt{n_i}\sim \sqrt{dn}$ as $n\rightarrow \infty$.

Proof

The function $\sqrt{x}$ is concave. This allows us to use Jensen’s inequality. We consider all weigths equal to 1 and obtain

$$\begin{aligned} \sqrt{\frac{\sum _{i=1}^{d} n_i}{d}}\ge \frac{\sum _{i=1}^{d} \sqrt{n_i}}{d} \end{aligned}$$

Thus $\sum _{i=1}^{d}\sqrt{n_i}\le \sqrt{dn}$.

Let $n=kd+r$ with $k,r\in \mathbb {N}, r< d$. Put $n_i= k+1$ for $i=1, \ldots , r$ and $n_i= k$ for $i=r + 1, \ldots , d$. We have $\sum _{i=1}^{d} n_i = kd+r = n$, and moreover $\sum _{i=1}^{d} \sqrt{n_i} \ge d \sqrt{k} = \sqrt{d(n-r)} \ge \sqrt{d(n-d)} \sim \sqrt{dn}$ as $n\rightarrow \infty$.$\square$

Lemma 4.3

Let $d\in \mathbb {N}, k\ge 1$. Let $(h(n))_{n\in \mathbb {N}}$ be a sequence such that $h(n)\sim C\frac{n}{L_k(n)}$ for some $C>0$. Then $\max _{n_1+\cdots +n_d=n}\sum _{i=1}^{d} h(n_i)\sim C\frac{n}{L_{k}(n)}$ as $n\rightarrow \infty$.

Proof

Let $\varepsilon >0$. By calculating the derivative and the second derivative of the function $\frac{x}{L_{k}(x)}$, it can be shown that there exists a positive constant $x_k$ such that $\frac{x}{L_{k}(x)}$ is positive, strictly monotone increasing and strictly concave on $(x_k, \infty )$. Moreover, assume that $x_k$ is large enough so that $|\frac{h(n)}{Cn/L_{k}(n)}-1|<\varepsilon$ for all $x_k\le n$. Let $M=\max (1, \max _{i\in [1,x_k]} h(i))$. Let $n>d(x_k+1)$ be arbitrary. Let $n_1\ge n_2\ge \cdots \ge n_d$ be such that $\sum _{i=1}^{d} n_i=n$. As $n>d(x_k+1)$, there exists a $1\le t\le d$ such that $n_i>x_k$ if and only if $i\le t$. We give an upper bound for $\sum _{i=1}^{d} h(n_i)$.

By using the trivial estimation $h(n_i)\le M$ for $i>t$, we have $\sum _{i=1}^{d} h(n_i)\le dM+\sum _{i=1}^{t} h(n_i)\le dM+\sum _{i=1}^{t} (1+\varepsilon )C\frac{n_i}{L_k(n_i)}$. Thus according to Jensen’s inequality $\sum _{i=1}^{d} h(n_i)\le dM+(1+\varepsilon )C\sum _{i=1}^{t} \frac{n_i}{L_k(n_i)}\le dM+(1+\varepsilon )Ct(\frac{1}{t}\sum _{i=1}^{t} \frac{n_i}{L_k(n_i)})\le dM+(1+\varepsilon )Ct\frac{n/t}{L_k(n/t)}= dM+(1+\varepsilon )C\frac{n}{L_k(n/t)}$.

As the $n_i$ were arbitrary, we have that $\max _{n_1+\cdots +n_d=n}(\sum _{i=1}^{d} h(n_i))\le dM+(1+\varepsilon )C\frac{n}{L_k(n/t)}\sim (1+\varepsilon )C\frac{n}{L_k(n)}$. A similar lower bound can be shown by setting all the $n_i$ so that the difference of any two of them is at most 1. Hence, the lower estimation that we obtain in this way is asymptotically $(1-\varepsilon )C\frac{n}{L_k(n)}$. As $\varepsilon >0$ was arbitrary, we have that $\max _{n_1+\cdots +n_d=n}(\sum _{i=1}^{d} h(n_i))\sim C\frac{n}{L_k(n)}$.

$\square$

Lemma 4.4

Let $\tau \in \mathbb {N}$, and let $1=d_1,d_2,\ldots , d_\tau$ be natural numbers. For $n\in \mathbb {N}$ let $w_{d_1,\ldots , d\tau }(n)$ be the number of tuples $(\alpha _1, \ldots , \alpha _\tau )$ of non-negative integers such that $\alpha _1d_1+\cdots +\alpha _{\tau }d_{\tau } = n$. Then $w_1(n)=1$ for all $n\in \mathbb {N}$ and for $\tau \ge 2$ we have $w_{d_1,\ldots , d\tau }(n)=\frac{1}{(\tau -1)!d_1d_2\cdots d_\tau }n^{\tau -1}+O(n^{\tau -2})$.

Proof

We prove the statement by induction on $\tau$. By definition, $w_1(n)=1$ for all $n\in \mathbb {N}$. Let $\tau =2$. Then we have ${\lfloor }\frac{n}{d_2}{\rfloor }+1$ choices for $\alpha _2$, and $\alpha _1$ is uniquely determined by $\alpha _2$. Thus $w_{1,d_2}(n)={\lfloor }\frac{n}{d_2}{\rfloor }+1=\frac{n}{d_2}+O(1)$.

Assume that $\tau \ge 3$, and that the assertion is true for $(\tau -1)$. We show that the statement holds for $\tau$. By rearranging the terms of $\alpha _1d_1+\cdots +\alpha _{\tau }d_{\tau } = n$ we obtain $\alpha _1d_1+\cdots +\alpha _{\tau -1}d_{\tau -1} = n-\alpha _{\tau }d_{\tau }$. Thus

$$\begin{aligned} w_{d_1,\ldots , d\tau }(n)&=\sum _{\alpha _\tau =0}^{{\lfloor }n/d_\tau {\rfloor }} w_{d_1,\ldots , d{\tau -1}}(n-\alpha _{\tau }d_{\tau }) \\&= \sum _{\alpha _\tau =0}^{{\lfloor }n/d_\tau {\rfloor }} \frac{1}{(\tau -2)!d_1d_2\cdots d_{\tau -1}}(n-\alpha _{\tau }d_{\tau })^{\tau -2} +O(n^{\tau -2}) \\&= \frac{d_{\tau }^{\tau -2}}{(\tau -2)!d_1d_2\cdots d_{\tau -1}}\sum _{\alpha _\tau =0}^{{\lfloor }n/d_\tau {\rfloor }} \left( \frac{n}{d_{\tau }}-\alpha _{\tau }\right) ^{\tau -2} +O(n^{\tau -2}) \\&= \frac{d_{\tau }^{\tau -2}}{(\tau -2)!d_1d_2\cdots d_{\tau -1}}\int \limits _{x=0}^{{\lfloor }n/d_\tau {\rfloor }}\! \left( \frac{n}{d_{\tau }}-x\right) ^{\tau -2}\,\mathrm {d}{x} +O(n^{\tau -2}) \\&= \frac{d_{\tau }^{\tau -2}}{(\tau -2)!d_1d_2\cdots d_{\tau -1}}\left( \frac{n}{d_{\tau }}\right) ^{\tau -1}/(\tau -1) +O(n^{\tau -2}) \\&= \frac{1}{(\tau -1)!d_1d_2\cdots d_\tau }n^{\tau -1}+O(n^{\tau -2}) \end{aligned}$$

$\square$

The following lemmas are used to determine the log-asymptotic behaviour of the generative- and fine spectra of $\mathcal {V}_{1,d}$ for $d\ge 2$.

Lemma 4.5

Let $a,b\in \mathbb {N}$. Then $\int \limits _{0}^{1} x^a(1-x)^b\;\mathrm {d}x = \frac{a!\cdot b!}{(a+b+1)!}$.

Proof

The expression $\int \limits _{0}^{1} x^a(1-x)^b\;\mathrm {d}x$ is clearly symmetric in a and b. If $b=0$, then $\int \limits _{0}^{1} x^a\;\mathrm {d}x= \frac{1}{a+1}$ holds, and by symmetry, the formula is also true when $a=0$.

By the rule of partial integration, we obtain

$$\begin{aligned}&(a+1)\int \limits _{0}^{1} x^{a}(1-x)^b\;\mathrm {d}x = \int \limits _{0}^{1} (x^{a+1})'(1-x)^b\;\mathrm {d}x \\&\quad = -\int \limits _{0}^{1} x^{a+1}((1-x)^b)'\;\mathrm {d}x = b\int \limits _{0}^{1} x^{a+1}(1-x)^{b-1}\;\mathrm {d}x \end{aligned}$$

Hence, the above formula is equivalent for pairs (a, b) and $(a',b')$ if $a+b=a'+b'$. $\square$

Lemma 4.6

Let $m\in \mathbb {N}^+$. For $x\ge 0$ define $S_{1,m}(x)= x^{m}$, and let

$$\begin{aligned} S_{i+1,m}(x)= \int \limits _{0}^{x} S_{i,m}(t)(x-t)^m\;\mathrm {d}t \end{aligned}$$

for all integers $i\ge 2$. Then

$$\begin{aligned} S_{i,m}(x)= \frac{(m!)^i}{((m+1)\cdot i-1)!}\cdot x^{(m+1)\cdot i-1} \end{aligned}$$

Proof

Induction on i with m fixed; the initial step $i=1$ holds by definition. Assume that the formula is true for $i\ge 1$, and let us show it for $i+1$. By using the induction hypothesis, the integral form of $S_{i,m}(x)$ transforms to

$$\begin{aligned} S_{i+1,m}(x)&= \int \limits _{0}^{x} S_{i,m}(t)(x-t)^m\;\mathrm {d}t\\&= \int \limits _{0}^{t} \frac{(m!)^i}{((m+1)\cdot i-1)!}\cdot t^{(m+1)\cdot i-1}(x-t)^m\;\mathrm {d}t \end{aligned}$$

By applying the linear substitution $y=t/x$ and Lemma 4.5 we obtain

$$\begin{aligned} S_{i+1,m}(x)&= \frac{(m!)^i}{((m+1)\cdot i-1)!}\cdot x^{(m+1)\cdot i-1+m+1}\int \limits _{0}^{1} y^{(m+1)\cdot i-1}(1-y)^m\;\mathrm {d}y\\&= \frac{(m!)^i}{((m+1)\cdot i-1)!}\cdot \frac{m!\cdot ((m+1)\cdot i-1)!}{(m\cdot (i+1) +i-1)!}\cdot \frac{x^{(m+1)\cdot (i+1) -1}}{(m+1)\cdot (i+1)-1} \\&= \frac{(m!)^{i+1}}{((m+1)\cdot (i+1)-1)!}\cdot x^{(m+1)\cdot (i+1) -1} \end{aligned}$$

$\square$

Lemma 4.7

Let $K\in \mathbb {R}^+$, $m, n\in \mathbb {N}^+$. Then for $\log n\le i\le n^{\frac{m+2}{m+3}}$ we have

$$\begin{aligned}&\log \left( \max _{i} \left( \frac{K^i}{i!}\frac{(m!)^i}{((m+1)\cdot i-1)!}\cdot n^{(m+1)\cdot i-1}\right) \right) \\&\quad = (m+2)\cdot \root m+2 \of {\frac{K\cdot m!}{(m+1)^{m+1}}}\cdot n^\frac{m+1}{m+2}+O(\log n) \end{aligned}$$

Proof

By Stirling’s formula, we have

$$\begin{aligned}&\log \left( \max _{i} \left( \frac{K^i}{i!}\frac{(m!)^i}{((m+1)\cdot i-1)!}\cdot n^{(m+1)\cdot i-1}\right) \right) \\&\quad = \max _{i} ( i\log K- i\log i +i + i\log m! - \log ((m+1)\cdot i - 1)! \\&\qquad + ((m+1)\cdot i - 1)\log n +O(\log i))\\&\quad = \max _{i} (i\cdot (\log K- \log i +1 + \log m!) - ((m+1)\cdot i - 1)\log ((m+1)\cdot i - 1)\\&\qquad +((m+1)\cdot i - 1)+ ((m+1)\cdot i - 1)\log n)+O(\log n) \\&\quad = \max _{i} (i\cdot (\log K- \log i +1 + \log m!) - (m+1)\cdot i \log ((m+1)\cdot i )\\&\qquad +(m+1)\cdot i + (m+1)\cdot i \cdot \log n)+O(\log n) \\&\quad = \max _{i} (i\cdot (\log K- \log i +1 + \log m!-(m+1)\log (m+1) - (m+1)\log i \\&\qquad + (m+1)+ (m+1)\log n)) + O(\log n) \\&\quad = \max _{i} (i\cdot (\log K + m + 2 + \log m!-(m+1)\log (m+1) - (m+2)\log i \\&\qquad + (m+1)\log n)) + O(\log n) \end{aligned}$$

We seek the maximum of this last expression in the interval $[\log n, n^\frac{m+2}{m+3}]$; the error this leads to has order of magnitude $O(\log n)$, as it is apparent from the derivative of the function we calculate now.

So let us define the following function

$$\begin{aligned} u(x)= & {} x\cdot (\log K + m + 2 + \log m!-(m+1)\log (m+1)\\&- (m+2)\log x + (m+1)\log n). \end{aligned}$$

Then the derivative of this funtion is

$$\begin{aligned} u'(x)= \log K + \log m!-(m+1)\log (m+1) - (m+2)\log x + (m+1)\log n. \end{aligned}$$

The equation $u'(x)=0$ has a unique solution, that is where u(x) attains its maximum, namely $x_0= \root m+2 \of {\frac{K\cdot m!}{(m+1)^{m+1}}}\cdot n^\frac{m+1}{m+2}$. Note that if n is large enough, then indeed $x_0\in [\log n, n^\frac{m+2}{m+3}]$, and that $u(x)= x(m+2+u'(x))$. Thus $u(x_0)= (m+2)x_0$, which is equivalent to the statement of the lemma.

$\square$

Lemma 4.8

Let $K\in \mathbb {R}^+$, $m,n\in \mathbb {N}^+$. Then for $0\le i\le n$ we have

$$\begin{aligned}&\log \left( \max _{i} \left( \frac{K^i}{i!}\cdot \sum _{r_1+\cdots +r_i= n}\prod _{j=1}^{i} r_j^m\right) \right) \\&\quad = (m+2)\cdot \root m+2 \of {\frac{K\cdot m!}{(m+1)^{m+1}}}\cdot n^\frac{m+1}{m+2} + O\left( n^\frac{m+1}{m+3}\right) \end{aligned}$$

Proof

Let $T_{i,m}(n) = \sum _{r_1+\cdots +r_i= n}\prod _{j=1}^{i} r_j^m$ be the discrete variant of the integral $S_{i,m}(x)$ defined in Lemma 4.6, and let $U_{i,m}(n) = \frac{K^i}{i!}\cdot T_{i,m}(n)$.

The number of terms in the sum $T_{i,m}(n)$ is $\left( {\begin{array}{c}n+i-1\\ i-1\end{array}}\right)$, and each term is at most $(\frac{n}{i})^{im}$ by the inequality between the arithmetic and geometric means. Thus putting $V_{i,m}(n):= \frac{K^i}{i!}\left( {\begin{array}{c}n+i-1\\ i-1\end{array}}\right) (\frac{n}{i})^{im}$, we have $U_{i,m}(n)\le V_{i,m}(n)$.

It is easy to see that   $\frac{V_{i+1,m}(n)}{V_{i,m}(n)} \le \frac{Kn^m(n+i)}{i^{m+1}(i+1)}$, which for $n^{\frac{m+2}{m+3}}<i\le n$ is at most $2K\cdot \frac{n^{m+1}}{i^{m+2}}< 2Kn^{-\frac{1}{m+3}} <1$ for n large enough. Hence, the sequence $a_i= \log (V_{i,m}(n))$ is strictly monotone decreasing on $[n^{\frac{m+2}{m+3}}, n]$, and its value at $i_0= \lceil n^{\frac{m+2}{m+3}}\rceil$ is at most

$$\begin{aligned}&\log \left( \frac{K^{i_0}}{i_0!}\cdot \left( {\begin{array}{c}n+i_0-1\\ i_0-1\end{array}}\right) \cdot \left( \frac{n}{i_0}\right) ^{i_0m}\right) \le \log \left( \frac{(2K)^{i_0}\cdot n^{i_0}\cdot n^{i_0m}}{i_0!\cdot i_0!\cdot i_0^{i_0m}}\right) \\&\quad \le i_0(\log (2Ke^2)+ (m+1)\log n - (m+2)\log i_0) \\&\quad \le i_0\left( \log (2Ke^2)+ (m+1)\log n - (m+2)\cdot \frac{m+2}{m+3}\cdot \log n\right) \\&\quad \le i_0\left( -\frac{1}{m+3}\cdot \log n +O(1)\right) <0 \end{aligned}$$

for n large enough.

Thus $U_{i,m}(n)\le V_{i,m}(n) \le V_{i_0,m}(n)< 1$ for n large enough for all $n^{\frac{m+2}{m+3}}<i\le n$.

For $i< \log n$ the trivial estimation $U_{i,m}(n)\le (2K)^{\log n}\cdot n^{\log n}\cdot n^{\log n}$ yields $\log (U_{i,m}(n))\le 2\log ^2 n + O(\log n) = O\left( n^\frac{m+1}{m+3}\right)$.

For $\log n\le i\le n^{\frac{m+2}{m+3}}$ we switch the sum $T_{i,m}(n) =\sum _{r_1+\cdots +r_i= n}\prod _{j=1}^{i} r_j^m =$ $\sum _{r_1= 0}^{n} \sum _{r_2= 0}^{n-r_1} \cdots \sum _{r_{i-1}= 0}^{n-r_1-\cdots - r_{i-2}} (r_1r_2\cdots r_{i-1})^m(n-r_1-\cdots - r_{i-1})^m$ to $S_{i,m}(n) =$ $\int \limits _{x_1=0}^{n} \int \limits _{x_2=0}^{n-x_1} \cdots \int \limits _{x_{i-1}=0}^{n-x_1-\cdots - x_{i-2}} (x_1x_2\cdots x_{i-1})^m(n-x_1-\cdots -x_{i-1}))^m\,\mathrm {d}x_{i-1}\cdots \mathrm {d}x_1$. This produces an error of order of magnitude $O\left( n^\frac{m+1}{m+3}\right)$, as $S_{i,m}(n-i)\le T_{i,m}(n)\le S_{i,m}(n+i)$, and by Lemma 4.6 we obtain

$$\begin{aligned}&\log \left( \frac{S_{i,m}(n+O(i))}{S_{i,m}(n)}\right) \log \left( \frac{n+O(i)}{n}\right) ^{(m+1)\cdot i -1} \\&\quad = \log \left( 1+O\left( \frac{i}{n}\right) \right) ^{(m+1)\cdot i -1}\\&\quad \le (m+1)\cdot n^{\frac{m+2}{m+3}}\cdot \log \left( 1+O\left( n^{-\frac{1}{m+3}}\right) \right) \\&\quad = (m+1)\cdot n^{\frac{m+2}{m+3}}\cdot O\left( n^{-\frac{1}{m+3}}\right) = O\left( n^\frac{m+1}{m+3}\right) \end{aligned}$$

The assertion then follows from Lemmas 4.6 and 4.7. $\square$

5 Fine spectra

5.1 The fine spectrum of the varieties $\mathcal {V}_k$

In [10] recursive formulas and asymptotic estimations were given for the number of n-element rooted trees of depth k. Those results directly imply the following.

Theorem 5.1

The sequences $f_k(n)$ satisfy the following asymptotic formulas.

• (1) $f_1(n)=1$ for all $n\in \mathbb {N}$.

• (2) $f_2(n)\sim \frac{1}{4\sqrt{3}n}\exp (\pi \sqrt{\frac{2}{3}}\sqrt{n})$.

• (3) $\log f_k(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)}$ for $k>2$.

5.2 The fine spectrum of the varieties $\mathcal {V}_{k,d}$

Theorem 5.2

The sequences $f_{k,d}(n)$ satisfy the following asymptotic formulas.

• (1) $\log f_{0,0}(n)\sim (\log \alpha )n$, where $\alpha \approx 2.95576$.

• (2) $f_{0,1}(n)=1$ for all $n\in \mathbb {N}$.

• (3) $f_{0,d}(n)\sim \frac{1}{(\tau (d)-1)!d^{\tau (d)/2}}\cdot n^{\tau (d)-1}$ for $d\ge 2$.

• (4) $f_{1,1}(n)=p(n)\sim \frac{1}{4\sqrt{3}n}\exp (\pi \sqrt{\frac{2}{3}}\sqrt{n})$.

• (5) $\log f_{1,d}(n)\sim \frac{(d+1)\cdot \root d+1 \of {\zeta (d)}}{d}\cdot n^\frac{d}{d+1}$ for $d\ge 2$, where $\zeta$ is the Riemann zeta function.

• (6) $\log f_{2,d}(n)\sim \frac{\pi ^2d}{6}\cdot \frac{n}{\log n}$ for $d\ge 1$.

• (7) $\log f_{k,d}(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-1}(n)}$ for $k\ge 3, d\ge 1$.

Proof

For the proof of item (1) see [6].

Item (2) is straightforward from the definition of $f_{0,1}(n)$.

For item (3), let $1=d_1, \ldots , d_{\tau (d)}$ be the positive divisors of d. An n-element algebra (A; u) in the variety $\mathcal {V}_{0,d}$ consists of disjoint cycles with size in $\{d_1, \ldots , d_{\tau (d)}\}$. Let us denote the number of cycles in (A; u) of size $d_i$ by $\alpha _i$. Then the isomorphism type of (A; u) is uniquely determined by the tuple $(\alpha _1, \ldots , \alpha _{\tau (d)})$. According to Lemma 4.4 the number of such tuples is $\frac{1}{(\tau (d)-1)!d_1 \cdots d_{\tau (d)}}n^{\tau (d)-1} +O(n^{\tau (d)-2})= \frac{1}{(\tau (d)-1)!d^{\tau (d)/2}}n^{\tau (d)-1} +O(n^{\tau (d)-2})$.

For item (4), observe that an n-element algebra (A; u) is in $\mathcal {V}_{1,1}$ if and only if it is the disjoint union of rooted trees of depth at most 1 with n vertices altogether, such that the edges are directed towards the root. A rooted tree with depth at most 1 is up to isomorphism uniquely determined by its size. Thus (A; u) is up to isomorphism uniquely determined by the partition of n corresponding to the multi-set of the sizes of the rooted trees.

We show item (5). The number of n-element directed, connected unicyclic graphs with cycle length d is asymptotically $\frac{1}{d}\left( {\begin{array}{c}n-1\\ d-1\end{array}}\right)$. Thus the number of n-element directed, connected unicyclic graphs with cycle length dividing d is asymptotically $\sum _{t|d} \frac{1}{t}\left( {\begin{array}{c}n-1\\ t-1\end{array}}\right) = (1+O(\frac{1}{n})) \frac{1}{d!} n^{d-1}$. Let $a_n = \sum _{m|n} \frac{1}{m} f_{1,d, con}(\frac{n}{m})$. Then $f_{1,d}(n)\le [x^n]\exp (\sum _{r=1}^\infty a_r x^r)$ by item (5) of Lemma 3.3, where $[x^n]H(x)$ denotes the coefficient of $x^n$ in the power series H(x). Furthermore, we have

$$\begin{aligned} a_n&\le \left( 1+O\left( \frac{1}{n}\right) \right) \sum _{m|n}\frac{1}{d!} \frac{1}{m} \left( \frac{n}{m}\right) ^{d-1} \\&\le \left( 1+O\left( \frac{1}{n}\right) \right) \frac{1}{d!} n^{d-1} \sum _{m=1}^\infty \left( \frac{1}{m}\right) ^{d}= \left( 1+O\left( \frac{1}{n}\right) \right) \frac{\zeta (d)}{d!} n^{d-1} \end{aligned}$$

Putting $K_0= \frac{\zeta (d)}{d!}, K= (1+\varepsilon )\cdot K_0$ and $m=d-1$, we obtain $a_n \le Kn^m$ for large enough n with arbitrary $\varepsilon >0$. Thus

$$\begin{aligned} \log (f_{1,d}(n))&\le \log \left( [x^n]\exp \left( \sum _{r=1}^\infty a_r x^r\right) \right) \\&\le (1+o(1)) \log \left( \sum _{i=1}^{n} \frac{K^i}{i!}\sum _{r_1+\cdots +r_i=n} \prod _{j=1}^{i} r_j^m\right) \\&\le (1+o(1)) \log \left( n\cdot \max _{i\in \{1, \ldots , n\}} \frac{K^i}{i!}\sum _{r_1+\cdots +r_i=n} \prod _{j=1}^{i} r_j^m\right) . \end{aligned}$$

Hence, Lemma 4.8 yields the asymptotic upper estimation

$$\begin{aligned}&\log \left( [x^n]\exp \left( \sum _{r=1}^\infty \frac{\zeta (d)}{d!} r^{d-1} x^r\right) \right) \sim (d+1)\cdot \root d+1 \of {\frac{\frac{\zeta (d)}{d!}\cdot (d-1)!}{d^d}}\cdot n^\frac{d}{d+1}\\&\quad = \frac{(d+1)\cdot \root d+1 \of {\zeta (d)}}{d}\cdot n^\frac{d}{d+1} \end{aligned}$$

for $\log f_{1,d}(n)$ as $\varepsilon \rightarrow 0$, as $K\rightarrow K_0$ then. The lower estimation can be obtained in a similar fashion. Let $\varepsilon >0$ be fixed, and choose $k\in \mathbb {N}$ such that $\sum _{m=1}^{k} \left( \frac{1}{m}\right) ^d\ge \zeta (d)-\varepsilon$. The only difference in the calculation compared to the upper estimation is that the inequality $(1-\varepsilon )\frac{\zeta (d)}{d!} n^{d-1}\le a_n$ does not hold for sufficiently large n, although for given $\varepsilon$, it is “often” true. The reason is that there are arbitrarily large numbers n with few divisors (e.g., primes), and for such an n we have $\sum _{m|n} \left( \frac{1}{m}\right) ^d< \zeta (d)-\varepsilon$. So instead of $[x^n]\exp \left( \sum _{r=1}^\infty a_rx^r\right)$, it is better to compute $[x^n]\exp \left( \frac{1}{1-x}\sum _{r=1}^\infty a_rx^r\right)$, to even out the numbers with few divisors. This modification clearly has no effect on the log-asymptotics, and as every number is close to a number n that is divisible by the first k numbers, we obtain a power series in the exponential whose n-th coefficient is asymptotically bigger than $(1-\varepsilon )\frac{\zeta (d)}{d!} n^{d-1}$. Hence, the lower estimation $(d+1)\cdot \root d+1 \of {\frac{(1-\varepsilon )\frac{\zeta (d)}{d!}\cdot (d-1)!}{d^d}}\cdot n^\frac{d}{d+1}\le \log f_{1,d}(n)$ holds for all $\varepsilon >0$ for sufficiently large n, which simplifies to the expression $\frac{(d+1)\cdot \root d+1 \of {(1-\varepsilon )\zeta (d)}}{d}\cdot n^\frac{d}{d+1}\le \log f_{1,d}(n)$ for large enough n.

We proceed with item (6). According to Lemma 3.3 item (4) and Theorem 5.1, $f_{2,1,{{\,\mathrm{con}\,}}}(n) = f_2(n)\sim \frac{1}{4n\sqrt{3}}\exp \left( \pi \sqrt{\frac{2n}{3}}\right)$. Lemma 3.3 item (3) yields $\frac{1}{d}\sum _{\mu _1+\cdots +\mu _d=n}\prod _{i=1}^{d} f_{2}(\mu _i)\le f_{2,d,{{\,\mathrm{con}\,}}}(n)\le \sum _{t|d}\sum _{\mu _1+\cdots +\mu _t=n}\prod _{i=1}^{t} f_{2}(\mu _i)$. Asymptotically there are at most $n^d$ terms in both the lower- and upper estimations, and according to Lemma 4.2 the logarithm of every term can be estimated by

$$\begin{aligned}&\log \left( \max _{\mu _1+\cdots +\mu _t=n}\prod _{i=1}^{t} f_{2}(\mu _i)\right) \\&\quad \le (1+o(1))\log \left( \max _{\mu _1+\cdots +\mu _t=n}\prod _{i=1}^{t} \frac{1}{4\mu _i\sqrt{3}}\exp \left( \pi \sqrt{\frac{2\mu _i}{3}}\right) \right) \\&\quad =O(t\log n)+ (1+o(1))\pi \sqrt{\frac{2}{3}}\max _{\mu _1+\cdots +\mu _t=n}\sum _{i=1}^{t} \sqrt{\mu _i}\\&\quad \le O(t\log n)+ (1+o(1))\pi \sqrt{\frac{2}{3}} \sqrt{tn} \\&\quad \le O(d\log n)+ (1+o(1))\pi \sqrt{\frac{2}{3}} \sqrt{dn}\sim \pi \sqrt{\frac{2}{3}} \sqrt{dn} \end{aligned}$$

Moreover, according to Lemma 4.2 the estimation is sharp when $t=d$ and the difference between any two of the $n_i$ is at most 1. Such a term appears in both the lower- and upper estimations. As $\log n^d$ is negligible to $\pi \sqrt{\frac{2}{3}} \sqrt{dn}$, it makes no difference in the log-asymptotic estimations if we calculate with the biggest term or the sum of the terms. Hence, both the lower- and upper estimations we obtained for $\log f_{2,d,{{\,\mathrm{con}\,}}}(n)$ are asymptotically $\pi \sqrt{\frac{2}{3}} \sqrt{d}\sqrt{n}$, and consequently, so is $\log f_{2,d,{{\,\mathrm{con}\,}}}(n)$. By Lemma 3.3 item (5) and Theorem 3.4 item (1) we have that $\log f_{2,d}(n)\sim \frac{\pi ^2d}{6}\cdot \frac{n}{\log n}$.

Finally, we show item (7). From Lemma 3.3 item (3) and Theorem 5.1 we obtain $\log f_{k,1,{{\,\mathrm{con}\,}}}(n) = \log f_k(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)}$. According to Lemma 3.3 item (4) we have $\frac{1}{d}\sum _{\mu _1+\cdots +\mu _d=n}\prod _{i=1}^{d} f_{k}(\mu _i)\le f_{k,d,{{\,\mathrm{con}\,}}}(n)\le \sum _{t|d}\sum _{\mu _1+\cdots +\mu _t=n}\prod _{i=1}^{t} f_{k}(\mu _i)$. Asymptotically there are at most $n^d$ terms in both the lower- and upper estimations, which contributes a negligible factor. Using Lemma 4.3 the logarithm of every summand can be estimated from above by

$$\begin{aligned} \log \left( \max _{\mu _1+\cdots +\mu _t=n}\prod _{i=1}^{t} f_{k}(\mu _i)\right)&= \max _{\mu _1+\cdots +\mu _t=n}\sum _{i=1}^{t} \log f_{k}(\mu _i) \\&\le (1+o(1))\max _{\mu _1+\cdots +\mu _t=n}\sum _{i=1}^{t} \frac{\pi ^2}{6}\cdot \frac{\mu _i}{L_{k-2}(\mu _i)} \\&\le (1+o(1))\frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)} \sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)} \end{aligned}$$

Moreover, according to the argument given in the proof of Lemmas 4.2 and 4.3 , the estimation is sharp when $t=d$ and the difference between any two of the $n_i$ is at most 1. Such a term appears in both the lower- and upper estimations. Hence, both the lower- and upper estimations we obtained for $\log f_{k,d,{{\,\mathrm{con}\,}}}(n)$ are asymptotically $\frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)}$, and consequently, so is $\log f_{k,d,{{\,\mathrm{con}\,}}}(n)$. By Lemma 3.3 item (5) and Theorem 3.4 item (2) we have that $\log f_{k,d}(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-1}(n)}$. $\square$

6 Generative spectra

6.1 The generative spectrum of the varieties $\mathcal {V}_k$

Theorem 6.1

The sequences $g_k(n)$ satisfy the following asymptotic formulas.

• (1) $g_1(n)=n+1$ for all $n\in \mathbb {N}$.

• (2) $g_2(n)\sim \frac{\sqrt{3}}{2\pi ^2}\exp \left( \pi \sqrt{\frac{2}{3}}\sqrt{n}\right)$.

• (3) $\log g_k(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)}$ for $k>2$.

Proof

Item (1) holds by definition.

For item (2), we first give an asymptotic estimation for $g_2^*(n)$. If (A; u) is an n-generated, but not $(n-1)$-generated algebra in $\mathcal {V}_2$, then $G_A$ is a rooted tree of depth at most 2 with n leaves. Let two leaves x and y be equivalent if $u(x)=u(y)$. Leaves x such that u(x) is the root form an equivalence class of $(n-i)$ elements, the others form a partition of an i-element set. The isomorphism type of (A; u) is uniquely determined by the number i and the partition of the i-element set. Thus $g_2^*(n)=\sum _{i=1}^{n} p(i)$. According to the Hardy-Ramanujan formula, $p(n)\sim \frac{1}{4\sqrt{3}n}\exp \left( \pi \sqrt{\frac{2}{3}}\sqrt{n}\right)$. By Lemma 4.1 we obtain $g_2^*(n)\sim \frac{1}{2\sqrt{2}\pi \sqrt{n}}\exp \left( \pi \sqrt{\frac{2}{3}}\sqrt{n}\right)$. Hence, $g_2(n)=\sum _{i=1}^{n} g_2^*(i)\sim \frac{\sqrt{3}}{2\pi ^2}\exp \left( \pi \sqrt{\frac{2}{3}}\sqrt{n}\right)$ by Lemma 4.1.

Finally, for item (3) it is enough to show that $\log g^*_k(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)}$ for $k>2$. We prove this estimation by induction on k. By Lemma 3.3 item (2) and Theorem 3.4 item (1), we obtain the result for $k=3$. Assume that the statement is true for some $k\ge 3$. Then by Lemma 3.3 item (2) and Theorem 3.4 item (2), the assertion holds for $k+1$, as well. $\square$

Corollary 6.2

The sequences $g^*_k(n)$ satisfy the asymptotic formulas

• (1) $g_2^*(n)\sim \frac{1}{2\sqrt{2}\pi \sqrt{n}}\exp \left( \pi \sqrt{\frac{2}{3}}\sqrt{n}\right)$ and

• (2) $\log g^*_k(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-2}(n)}$ for $k>2$.

6.2 The generative spectrum of the varieties $\mathcal {V}_{k,d}$

Theorem 6.3

The sequences $g_{k,d}(n)$ satisfy the following asymptotic formulas.

• (1) $g_{0,d}(n)= \left( {\begin{array}{c}\tau (d)+n\\ n\end{array}}\right) - 1\sim \frac{1}{\tau (d)!}n^{\tau (d)}$ for $d\ge 1$.

• (2) $g_{1,1}(n)\sim \frac{\sqrt{3}}{2\pi ^2}\exp \left( \pi \sqrt{\frac{2}{3}}\sqrt{n}\right)$.

• (3) $\log g_{1,d}(n)\sim \frac{(d+1)\cdot \root d+1 \of {\zeta (d)}}{d}\cdot n^\frac{d}{d+1}$ for $d\ge 2$, where $\zeta$ is the Riemann zeta function.

• (4) $\log g_{2,d}(n)\sim \frac{\pi ^2d}{6}\cdot \frac{n}{\log n}$ for $d\ge 1$.

• (5) $\log g_{k,d}(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-1}(n)}$ for $k\ge 3,d\ge 1$.

Proof

For item (1) observe that an algebra (A; u) in $\mathcal {V}_{0,d}$ is n-generated if and only if (A; u) consists of at most n disjoint cycles. The length of a cycle can be any divisor of d. Thus up to isomorphism (A; u) is uniquely determined by the multi-set of i numbers, with $i\le n$, consisting of the sizes of the cycles in (A; u), and these i numbers can be chosen from a $\tau (d)$-element set. Hence, $g_{0,d}(n)=\sum _{i=1}^{n} \left( {\begin{array}{c}\tau (d)+i-1\\ i\end{array}}\right) = \left( {\begin{array}{c}\tau (d)+n\\ n\end{array}}\right) - 1$.

For item (2) observe that there is a bijection between $\mathcal {V}_{2}$ and $\mathcal {V}_{1,1}$: if we omit the root of an algebra in $\mathcal {V}_{2}$ then we obtain an algebra in $\mathcal {V}_{1,1}$. Moreover, this bijection maps n-generated algebras in $\mathcal {V}_{2}$ to n-generated algebras in $\mathcal {V}_{1,1}$. Thus $g_{1,1}(n)=g_2(n)$, and we are done by Theorem 6.1 item (2).

The proof of item (3) is analogous to that of Theorem 5.2 item (5).

For items (4) and (5) it is enough to show that $\log g^*_{2,d}(n)\sim \frac{\pi ^2d}{6}\cdot \frac{n}{\log n}$ for $d\ge 1$ and $\log g^*_{k,d}(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-1}(n)}$ for $k\ge 3, d\ge 1$. By comparing Theorem 5.1 and Corollary  we obtain that $\log f_k(n)\sim \log g^*_{k}(n)$ for $k\ge 2$. In the statement of Lemma 3.3 items (3), (4) and (5) are analogous to items (6), (7) and (8). Hence, the proofs of the desired log-asymptotic estimations $\log g^*_{2,d}(n)\sim \frac{\pi ^2d}{6}\cdot \frac{n}{\log n}$ for $d\ge 1$ and $\log g^*_{k,d}(n)\sim \frac{\pi ^2}{6}\cdot \frac{n}{L_{k-1}(n)}$ for $k\ge 3, d\ge 1$ are also analogous to the proofs of items (6) and (7) of Theorem 5.2. $\square$

Many of our results determine the $\log$-asymptotic behaviour of the fine- and generative spectra of varieties of monounary algebras. This can be considered as a first step towards more refined asymptotic formulas. As we mentioned in the introduction, our results provide infinitely many examples of spectra with $\log$-asymptotic behaviour strictly between polynomials and exponential functions. It would be particularly interesting to find asymptotic formulas for $f_{1,d}(n)$ and $g_{1,d}(n)$, the two series that turned out to be the most challenging to deal with in the present paper. We are certain that such a result would require more advanced techniques, for example the saddle point method, which was effectively used in similar problems in the past, see [3].