The fine- and generative spectra of varieties of monounary algebras

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.


Introduction
For a variety of algebras V let g V (n) denote the number of n-generated algebras in V, and let f V (n) denote the number of n-element algebras in V up to isomorphism. The sequences (g V (n)) n∈N and (f V (n)) n∈N are called the generative spectrum and the fine spectrum of 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 V of groups is nilpotent if and only if g V (n) is at most polynomial, and a finite ring R generates a variety with at most exponential 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 .

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 V k,d are defined by the equation u k (x) = u k+d (x), for k ≥ 0, d ≥ 1. An algebra (A; u) is in 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 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 V 0,0 defined by the equation x = x.
As there are infinitely many n-generated algebras in V 0,0 for all n, the generative spectrum of this variety is not defined. In view of [6], the fine spectrum of V 0,0 is (M n ) n∈N and log M n ∼ n log α, where α ≈ 2.95576. • The varieties V k are defined by the equation u k (x) = u k (y), for k ≥ 1.
The classes V k consist of connected monounary algebras. If (A; u) ∈ 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) ∈ 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 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 V k was determined in [10]. The logasymptotic 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.

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 . . . log x. The exponential function e x is denoted by exp(x). The number of positive divisors of n is denoted by τ (n).
We use the symbols ∼, o(.), O(.) in the standard way. Given two series a, b : N → R with b n = 0 for all but finitely many n, we put a n ∼ b n if lim n→∞ an bn = 1, a n = o(b n ) if lim n→∞ an bn = 0, and a n = O(b n ) if an bn is bounded. Furthermore, the expression "log-asymptotic behaviour of a n " refers to an asymptotic estimation of the sequence (log a n ) n∈N , i.e., a sequence (b n ) n∈N with log a n ∼ b n . Note that finding such a sequence (b n ) n∈N is usually less demanding than finding a sequence (c n ) n∈N with a n ∼ 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 → ∞, then clearly log a n ∼ b n for b n = log c n .

Definition 3.2.
• For k ≥ 0, f k (n) is the number of non-isomorphic nelement algebras in V k , which equals to the number of n-element rooted trees of depth at most k. The generating function of the sequence 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 is denoted by G * k (x) = ∞ n=1 g * k (n)x n . • For k ≥ 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 V k . The generating function of the sequence (g k (n)) ∞ n=1 is denoted by is the number of non-isomorphic connected n-element algebras in 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,con (n)) ∞ n=1 is denoted by is the number of non-isomorphic connected n-generated but not (n − 1)-generated algebras in 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, 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. (1) is shown in [10], see Theorem 2.2. The proof of item (2) is analogous.

Lemma 3.3. The formal power series defined in Definition 3.2 satisfy the following formulas coefficient-wise for all integers
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 ≤ i ≤ n let μ 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 . According to the generalised binomial theorem, for every |x| Finally, the proofs of items (4) and (7) are similar, thus we only show item (4). Let (A; u) be a connected algebra in V k,d such that the length of its cycle is t. Then t|d. Let r 1 , . . . , r t be an enumeration of the elements of the cycle of (A; u) such that u(r 1 ) = r 2 , . . . , 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 , . . . , 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 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 , . . . , 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 V k,d . Hence, the above argument shows that 1 which is the n-th coefficient in the power series (F k,1,con (x)) t .
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. (1) If log a n ∼ C √ n for some C > 0, then log b n ∼ C 2 4 n log n . (2) For k ≥ 1, if log a n ∼ C n L k (n) for some C > 0, then log b n ∼ C n L k+1 (n) .

Auxiliary computations
Proof. As a n → ∞ and a n ∼ Kn s exp( Proof. The function √ x is concave. This allows us to use Jensen's inequality. We consider all weigths equal to 1 and obtain Proof. Let ε > 0. By calculating the derivative and the second derivative of the function x L k (x) , it can be shown that there exists a positive constant x k such that x L k (x) is positive, strictly monotone increasing and strictly concave on (x k , ∞). Moreover, assume that x k is large enough so that | h(n)

Lemma 4.4. Let
Proof. We prove the statement by induction on τ . By definition, w 1 (n) = 1 for all n ∈ N. Let τ = 2. Then we have n d2 + 1 choices for α 2 , and α 1 is uniquely determined by α 2 . Thus w 1,d2 (n) = n d2 + 1 = n d2 + O (1). Assume that τ ≥ 3, and that the assertion is true for (τ −1). We show that the statement holds for τ . By rearranging the terms of The following lemmas are used to determine the log-asymptotic behaviour of the generative-and fine spectra of V 1,d for d ≥ 2.
Proof. The expression x a dx = 1 a+1 holds, and by symmetry, the formula is also true when a = 0. By the rule of partial integration, we obtain (a + 1) Hence, the above formula is equivalent for pairs (a, b) and (a , b ) Proof. Induction on i with m fixed; the initial step i = 1 holds by definition. Assume that the formula is true for i ≥ 1, and let us show it for i + 1. By using the induction hypothesis, the integral form of S i,m (x) transforms to By applying the linear substitution y = t/x and Lemma 4.5 we obtain Proof. By Stirling's formula, we have  The number of terms in the sum T i,m (n) is n+i−1 i−1 , and each term is at most ( n i ) im by the inequality between the arithmetic and geometric means.
For i < log n the trivial estimation U i,m (n) ≤ (2K) log n · n log n · n log n yields log(U i,m (n)) ≤ 2 log This produces an error of order of magnitude O n m+1 m+3 , as S i,m (n − i) ≤ T i,m (n) ≤ S i,m (n + i), and by Lemma 4.6 we obtain The assertion then follows from Lemmas 4.6 and 4.7.

The fine spectrum of the varieties 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.

The fine spectrum of the varieties V k,d
Theorem 5.2. The sequences f k,d (n) satisfy the following asymptotic formulas.
Item (2) (4), observe that an n-element algebra (A; u) is in 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