Abstract
A tournament on a graph is an orientation of its edges. The score sequence lists the indegrees in nondecreasing order. Works by Winston and Kleitman (J Comb Theory Ser A 35(2):208–230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197–206, 2000) showed that the number \(S_n\) of score sequences on the complete graph \(K_n\) satisfies \(S_n=\Theta (4^n/n^{5/2})\). By combining a recent recurrence relation for \(S_n\) in terms of the Erdős–Ginzburg–Ziv numbers \(N_n\) with the limit theory for discrete infinitely divisible distributions, we observe that \(n^{5/2}S_n/4^n\rightarrow e^\lambda /2\sqrt{\pi }\), where \(\lambda =\sum _{k=1}^\infty N_k/k4^k\). This limit agrees numerically with the asymptotics of \(S_n\) conjectured by Takács (J Stat Plan Inference 14(1):123–142, 1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters \(r=2\) and \(p=e^{\lambda }\).
1 Introduction
A tournament on a graph G is an orientation of its edges. Informally, each vertex is a team and each pair of vertices joined by an edge plays a game. Afterwards, the edge is directed towards the winner. The score sequence lists the total number of wins by each team in nondecreasing order. For a survey, see, e.g., Harary and Moser [16] and Moon [26].
Let \(S_n\) denote the number of score sequences on the complete graph \(K_n\). Calculations related to \(S_n\) appear in the literature as early as MacMahon [24]. The investigation of its asymptotics began with the unpublished work of Erdős and Moser (see [26]) which showed \(c/n^{9/2}\le S_n/4^n\le C/n^{3/2}\). Winston and Kleitman [41] (cf. Kleitman [21]) proved \(S_n/4^n\ge c/n^{5/2}\), and argued that an upper bound of the same order follows, supposing that the largest qCatalan number is \(O(4^n/n^3)\). This fact was finally verified by Kim and Pittel [20], and hence \(S_n=\Theta (4^n/n^{5/2})\), as conjectured by Moser [28].
In this note, we locate the precise asymptotics. Let \(N_n\) denote the number of subsets of \([2n1]=\{1,2,\ldots 2n1\}\) of size n that sum to a multiple of n (see Sect. 1.1 below).
Theorem 1
As \(n\rightarrow \infty \), we have that
Here, as usual, \(f(n)\sim g(n)\) means that \(f(n)/g(n)\rightarrow 1\), as \(n\rightarrow \infty \). The quantity in the exponential will appear often throughout, so we put
We note that, from a geometric point of view, \(S_n\) is the number of nondecreasing lattice points in the permutahedron \(\Pi _{n1}\), the convex hull of all permutations of the vector \(v_n=(0,1,\ldots ,n1)\), see Fig. 1. Note that \(v_n\) is the score sequence for the tournament on \(K_n\) in which each team i wins against all teams \(j<i\) of smaller index (and loses against all other teams \(j>i\)). This polytope is equivalently obtained as the graphical zonotope of \(K_n\), more specifically, the (translated) Minkowski sum of line segments \(v_n+\sum _{i<j}[0,e_ie_j]\), where \(e_i\) are the standard basis vectors. See, e.g., Rado [30], Landau [23], Stanley [35], and Ziegler [44] for more details on these connections. See also, e.g., Postnikov [29] (and references therein) for results on counting lattice points in the permutahedron and other polytopes. Let us also note that \(S_n\) is the number of lattice points in the partitioned permutahedron \(P_{A_{n1}}([n1])\) in the recent work of Horiguchi et al. [18].
An intriguing connection between \(S_n\) and \(N_n\) was conjectured by Hanna [32] and recently proved by Claesson et al. [6, Corollary 12]. As it turns out,
This follows as a corollary of the main result in [6, Theorem 4] which establishes a relationship between generating functions related to the numbers \(S_n\), \(N_n\) and “pointed” score sequences (with a distinguished index). Another consequence of their main result is a beautiful connection between score sequences and the Catalan numbers, see [6, Corollary 11]. See also Sect. 3.3 below for some probabilistic intuition, via renewal theory, for the recursion (2).
The potential for a relationship between \(S_n\) and \(N_n\) is perhaps not so obvious at first blush, however, let us remark here that, as observed in [41, p. 212], each score sequence \(s\in {\mathbb Z}^n\) is associated with a subset \(\{s_1+1,s_2+2,\ldots ,s_n+n\}\) of \(\{1,2,\ldots ,2n1\}\), which sums to \(n^2\). Informally, if the sequence s is drawn as a “bar graph” consisting of upward and rightward unit steps, then this subset contains the “times” at which the walk takes a rightward step, thus “completing a bar.”
The relationship (2) implies that the sequence \(\{S_0,S_1,\ldots \}\) is infinitely divisible (see Sect. 2 below). The theory of infinitely divisible distributions began with de Finetti [10]. See, e.g., Wright [42], Hawkes and Jenkins [17], van Harn [40], Embrechts and Hawkes [12] and Steutel and van Harn [37] for background and results in the discrete case which are pertinent to the current article. Often, when two sequences are related in this way, asymptotic information can be transferred between them. The purpose of this note is to point out that the asymptotics of \(S_n\) can be obtained from those of \(N_n\), via the limit theory developed in [12, 17]. See Sect. 3 for the proof of Theorem 1.
1.1 Takács’ Conjecture
A wellknown result of Erdős et al. [13] states that every set I of \(2n1\) integers has a subset of size n which sums to a multiple of n. In the special case \(I=\{1,2,\ldots ,2n1\}\) that the integers are consecutive, the number \(N_n\) of such subsets is equal to
It appears that this formula is, in fact, but a special case of a result by von Sterneck from the early 1900s. See the discussion before Lemma 3.1 in the recent work of Konvalinka et al. [22], which cites Ramanathan [31]. See also the recent, independent proofs by Alekseyev [1] and Chern [4].
In Sect. 3.1, we use (3) together with Theorem 1 to obtain the following asymptotics for \(S_n\), in line with those conjectured by Takács [39, p. 136].
Corollary 2
As \(n\rightarrow \infty \), we have that
1.2 Irreducible Subscores
In a natural way, any tournament (or score sequence) can be decomposed into a series of irreducible parts (see, e.g., Moon and Moser [27] and Wright [43]). Landau [23] showed that a sequence \(s\in {\mathbb Z}^n\) is a tournament score sequence on \(K_n\) if and only if \(\sum _{i=1}^k s_n\ge {k\atopwithdelims ()2}\) with equality when \(k=n\). In other words, if and only if s is majorized (see, e.g., Marshall et al. [25]) by \(v_n=(0,1,\ldots ,n1)\). We let \(S_{n,m}\) denote the number of score sequences with exactly m irreducible subscores, that is, points k at which \(\sum _{i=1}^k s_i={k\atopwithdelims ()2}\) takes the smallest possible value. Hence \(S_n=\sum _{m=1}^n S_{n,m}\).
In Sect. 2, we show that \(\{S_0,S_1,\ldots \}\) is a renewal sequence (see, e.g., Feller [14, Sect. XIII]). In Sect. 3.2, we use this to deduce that, asymptotically, the exponential term in (1) has the following probabilistic interpretation.
Corollary 3
Let \({\mathcal I}_n\) denote the number of irreducible subscores in a uniformly random tournament score sequence on \(K_n\). Then, as \(n\rightarrow \infty \), the expected inverse number of such subscores satisfies
We note that
is the harmonic renewal measure (see, e.g, Greenwood et al. [15]) associated with the measure \(\mu _n=S_{n,1}/4^n\) (which has total measure \(1e^{\lambda }\)). Moreover, \(\beta _n\) is the Lévy measure (see, e.g., [12]) associated with the infinitely divisible probability distribution \(p_n=e^{\lambda }S_n/4^n\), which we call the tournament distribution. See Fig. 2. See the discussion at the end of Sect. 3.2 for more details.
1.3 Strong Score Sequences
Let \(S_{n,1}\) denote the number of strong (irreducible) score sequences. The reason for the name is that a score sequence s of a strongly connected tournament (in which each pair of vertices are in a directed cycle) is irreducible, that is, \(\sum _{i=1}^k s_i>{k\atopwithdelims ()2}\) for all \(k<n\). In other words, s is strictly majorized by \(v_n\). See, e.g., [16, Theorem 9].
Combining the “reverse renewal theorem” in Alexander and Berger [2] with the results above, we obtain the following.
Corollary 4
As \(n\rightarrow \infty \), we have that
and so
These asymptotics agree with those conjectured by Kotesovec [33], as discussed recently in Stockmeyer [38, Sect. 6]. Note that, for all sufficiently large n, more than half of all score sequences are strong.
1.4 Limiting Distribution
With these asymptotics at hand, we can then apply the powerful local limit theory of Chover et al. [5] to, more generally, obtain the asymptotic distribution of the number of irreducible subscores in a random score sequence.
Corollary 5
Let \({\mathcal I}_n\) denote the number of irreducible subscores in a uniformly random tournament score sequence on \(K_n\). Then, we have the following convergence in distribution,
where \({\mathcal N}_\lambda \) is a negative binomial (Pascal) random variable with parameters \(r=2\) and \(p=e^{\lambda }\). That is, for any \(m\ge 1\),
as \(n\rightarrow \infty \). In particular,
Note that this is in agreement with the fact, by Corollary 3, that
We also note that the mean
and variance
Equivalently, \({\mathcal N}_\lambda \) can be expressed as a compound random variable \(\sum _{i=1}^N X_i\), where N is Poisson with rate \(2\log (1e^{\lambda })\) and the \(X_i\) are independent and identically logarithmically distributed as
Hence the limiting distribution \({\mathcal N}_\lambda \) is infinitely divisible (see, e.g., [14, Sect. XII.2, p. 290]). See Fig. 3.
2 Infinite Divisibility
Following [17], we call a positive sequence \(\{1=a_0,a_1,\ldots \}\) infinitely divisible if for any integer \(r\ge 1\) there is a nonnegative sequence \(\{b_0,b_1,\dots \}\) such that \(a_n=b_n^{*r}\), where \(b_n^{*r}\) is the rth convolution power, defined inductively by \(b_n^{*1}=b_n\) and \(b_n^{*r}=\sum _{k=0}^n b_{k}b_{nk}^{*(r1)}\), for \(r>1\). In [17, Theorem 2.1] (cf., e.g., Katti [19], Steutel [36] and van Harn [40]) it is shown that a positive sequence \(\{1=a_0,a_1,\ldots \}\) is infinitely divisible if and only if
for some other nonnegative sequence \(\{0={\hat{a}}_0,{\hat{a}}_1,\ldots \}\). We call \({\hat{a}}_n\) the log transform of \(a_n\), since their generating functions \(A(x)=\sum _{n=0}^\infty a_n x^n\) and \({\hat{A}}(x)=\sum _{n=1}^\infty {\hat{a}}_n x^n\) satisfy
and so,
As discussed above, it was recently proved in [6] that the log transform of \(S_n\) is \({\hat{S}}_n=N_n\). We note that, in [6], \({\hat{A}}(x)\) is called the “logarithmic pointing” of A(x) and it is observed that, when (4) holds, a certain “closed formula” for \(a_n\) can be obtained in terms of the \({\hat{a}}_k\). Using this, see [6, Corollary 14], it is shown that
where \(\textrm{Sym}(n)\) is the symmetric group on \(\{1,2,\ldots ,n\}\) and \(C(\pi )\) is the sequence of cycle lengths in \(\pi \). However, while this gives a “closed formula” for \(S_n\), it is not clear to us if the asymptotics of \(S_n\) can be easily obtained from this formula.
3 Proofs
Our main result follows quite simply by the limit theorems in [12, 17].
Proof of Theorem 1
Put
Then, by (2), we obtain
In [12] it is shown that, for sequences of this form, the asymptotics of \(\beta _n\) can be transferred to \(\alpha _n\), under the following conditions.
Theorem 6
([12, Theorem 1]) Suppose that \(\{1=\alpha _0,\alpha _1,\ldots \}\) and \(\{\beta _1,\beta _2,\ldots \}\) are positive sequences satisfying (6) and that \(\lambda =\sum _{k=1}^\infty \beta _k<\infty \). Put \(\beta _0=0\). Then, if \(\beta _{n+1}\sim \beta _n\) and \(\beta _n^{*2}\sim 2\lambda \beta _n\), it follows that \(\alpha _n\sim e^\lambda \beta _n\).
In order to apply this result, we first note that, in this instance, we have by (3) (see Lemma 11 in Appendix 1 below) that
The condition \(\beta _{n+1}\sim \beta _n\) is clear. Next, we note that
and
Hence the theorem applies, and Theorem 1 follows. \(\square \)
Note that, in our application of Theorem 6, the infinitely divisible distribution at play is
which we call the tournament distribution. See Fig. 2 above. The fact that \(p_n\) is a probability distribution follows by (5) and that \({\hat{S}}_n=N_n\) by (2). Recall that (see, e.g., [14, Sect. XII.2, p. 290]) an infinitely divisible distribution on the nonnegative integers is compound Poisson, that is, distributed as a random sum \(\sum _{i=1}^M X_i\) of Poisson M many independent and identically distributed random variables \(X_i\), which are moreover independent of M. In this instance, \(\textbf{P}(X_i=n)=\beta _n/\lambda \) and M is Poisson with rate \(\lambda \).
Let us also observe that, alternatively, Theorem 1 can be proved from (2) using [17, Theorem 3.1]. Indeed, put
Then note that
and that, by (3), \(b_n\) is regularly varying with index \(p=3/2\). See [17] for definitions and more details.
3.1 The Constant
In this section, we obtain a numerical approximation to the constant factor in the leading order asymptotics of \(S_n\), verifying the asymptotics conjectured in [39, p. 136].
Proof of Corollary 2
By Theorem 1, we have that
Next, we use the technical estimates in Lemma 11, proved in Appendix 1 below. By (16), it follows that
The values \(N_k\), for all \(k\le 1669\), have been listed by Manyama and Noe [34]. Using these, the sum
can be computed. Then, applying (17), we find that
Hence
In particular,
as conjectured by Takács. \(\square \)
Kotesovec [32] reported (without explanation) the numerical approximation
We note that (9) confirms the first 8 decimal digits in (10).
3.2 The Exponential Term
Next, we obtain a probabilistic interpretation (Corollary 3 above) of the exponential term in (1), which is more closely related (on the face of it) to score sequences. To this end, we first prove the following result, relating \(N_n\) and \(S_{n,m}\) (see Sects. 1 and 1.2 above for the definitions). This follows by the recursion (2) and a general fact (Lemma 8 below) about renewal sequences, and is proved below (after a short example and the proof of Lemma 8).
Proposition 7
For all \(n\ge 1\), we have
For example, when \(n=6\), we have \(S_{6,1}=7\), \(S_{6,2}=7\), \(S_{6,3}=3\), \(S_{6,4}=4\), \(S_{6,5}=0\) and \(S_{6,6}=1\). Hence
On the other hand, we obtain the same total sum by computing (where \(d=\textrm{gcd}(n,k)\))
Following [17], we call a positive sequence \(\{1=a_0,a_1,\ldots \}\) a renewal sequence if, for some other sequence \(\{0=b_0,b_1,\ldots \}\), we have that
where \(A(x)=\sum _{n=0}^\infty a_nx^n\) and \(B(x)=\sum _{n=1}^\infty b_nx^n\) are their generating functions. As noted in [17] (p. 66), any such sequence is infinitely divisible. More specifically, let us note the following.
Lemma 8
([17]) Suppose that \(\{1=a_0,a_1,\ldots \}\) is a renewal sequence satisfying (11). Then \(\{1=a_0,a_1,\ldots \}\) is infinitely divisible and the log transform of \(a_n\) is
Proof
Taking logs on both sides of (11) and then differentiating, we find that
Then, comparing coefficients,
as required. \(\square \)
Proof of Proposition 7
Setting \(S_0=1\) and \(S_{0,m}=0\), we haves

(1)
\(S_n=\sum _{m=1}^n S_{n,m}\), for \(n\ge 1\), and

(2)
\(S_{n,m}=\sum _{k=1}^n S_{k,1}S_{nk,m1}\), for \(1<m\le n\).
Therefore, if we let \(S(x)=\sum _n S_{n}x^n\) and \(S_m(x)=\sum _n S_{n,m}x^n\) denote their generating functions, then by induction
and so
Hence, the result follows by (2) and Lemma 8, noting that \(S_{n,m}=[x^n][S_1(x)^m]\). \(\square \)
Proof of Corollary 3
By Theorem 1 and Proposition 7,
Using the numerics (8) (in the proof of Corollary 2 above) it follows that
as claimed. \(\square \)
Let us note here that by (6), (7) and Proposition 7, we see that the Lévy measure (see, e.g., [12])
corresponding to the tournament distribution \(p_n\), is the harmonic renewal measure (see, e.g, [15])
associated with the measure \(\mu _n=S_{n,1}/4^n\).
3.3 Probabilistic Intuition for (2)
Using (12) and a bijection in [6], we can see that (2) holds as follows. Indeed, by Lemma 8 and (12), we have that
where
Consider selecting a uniformly random score sequence s of length n and an independent and uniformly random integer \(\ell \in [n]\). By (12), the probability that the final subscore of s is of length \(\ge \ell \) is equal to
On the other hand, in [6, Lemma 10] a bijection is constructed between subsets \(I\subset [2n1]\) of size n contributing to \(N_n\) and pairs \((s,\ell )\), where s is a score sequence of length n with final subscore of some length k and \(\ell \in [k]\). Specifically, for each such score sequence s, with coordinates \(s_1\le \cdots \le s_n\), and each \(\ell \in [k]\), let \(I_s=\{y_1,\ldots , y_n\}\) be the subset of \([2n1]\) with \(y_i=x_i+i\), where \(x_1\le \cdots \le x_n\) is the nondecreasing rearrangement of the integers \(s_in+\ell \) mod n. Therefore, the above probability is equivalently calculated as \(N_n/nS_n\). Hence \({\hat{S}}_n=N_n\), and (2) follows.
For example, when \(n=4\), note that \(S_4=4\) and \(N_4=9\). The final subscore in 0123, 1113, 0222 and 1122 is of length 1, 1, 3 and 4. The subset associated with 0123 and \(\ell =1\) is 1357. The subset associated with 1113 and \(\ell =1\) is 1456. The subsets associated with 0222 and \(\ell =1,2,3\) are 2567, 1236 and 2347. The subsets associated with 1122 and \(\ell =1,2,3,4\) are 3467, 1267, 1245 and 2356.
3.4 Strong Scores
In this section, we prove Corollary 4. See, e.g., [14, Sect. XIII] for background on discrete renewal theory. Following the notation there, we put
Then by (12)
where \(U(s)=\sum _{n=0}^\infty u_n s^n\) and \(F(s)=\sum _{n=1}^\infty f_n s^n\) are the associated generating functions. Note that, by (2) and the numerics in Corollary 3, we have that
Hence \(F(1)<1\). We extend the “defective” distribution \(\{f_n:n\ge 1\}\) to a probability distribution by setting \(f_\infty =1F(1)>0\).
This relationship (in the notation of [2]) corresponds to a “transient” renewal process of arrival times
with independent and identically distributed interarrival times \(\Delta \tau _i\) distributed as \(\textbf{P}(\tau _1=n)=f_n\) and \(\textbf{P}(\tau _1=\infty )=f_\infty \). By (13) it follows that \(u_n=\textbf{P}(n\in \tau )\). Such a renewal process almost surely terminates (that is, eventually some \(\Delta \tau _i=\infty \)) after some finite amount of time, and in that sense is transient.
There is interest in the asymptotics of sequences \(u_n\) and \(f_n\) related by (12). See, e.g., the series of works by de Bruijn and Erdős [7,8,9]. Most results in the literature give information about the asymptotics of \(u_n\), assuming certain conditions on the regularity of \(f_n\) (which, as discussed in [2], can in practice be difficult to verify). However, in our current situation, having already established the asymptotics of \(u_n\), we want to go in the other direction. To this end, the following “reverse renewal theorem” by Alexander and Berger is just what we need.
Theorem 9
([2, Theorem 1.4]) If \(u_n\) is regularly varying and \(\tau \) is transient, then \(f_n/u_n\rightarrow f_\infty ^2\) as \(n\rightarrow \infty \).
As discussed in [2], this result can also be proved using techniques from Banach algebra [5, 11]. However, see [2] for a short probabilistic proof.
Proof of Corollary 4
By Theorem 1 and Corollary 2, the sequence \(u_n\) is regularly varying with index \(5/2\). In fact, \(n^{5/2}u_n\) converges. Therefore, applying Theorem 9 and (14), it follows that
and so by Corollary 2
as required. \(\square \)
3.5 Limiting Distribution
Finally, we prove Corollary 5, which extends the result of Corollary 4. To this end, we invoke the following special case of a result by Chover, Ney and Wainger [5] (cf. [3, Chapter IV]).
Theorem 10
([5, Theorem 1]) Let \(\{\mu _1,\mu _2,\ldots \}\) be a probability measure for which \(\mu _{n+1}/\mu _n\rightarrow 1\), as \(n\rightarrow \infty \), and so that, for all sufficiently large n,
for some constant C. Then, for any \(m\ge 2\), we have that \(\mu _n^{*m}/\mu _n\rightarrow m\), as \(n\rightarrow \infty \).
Proof of Corollary 5
is a probability measure. By Corollary 4, we have that \(\mu _{n+1}/\mu _n\rightarrow 1\). Moreover,
and so condition (15) is clear. Hence, by Theorem 10,
Therefore, by Corollary 4,
as claimed. \(\square \)
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We thank Jim Pitman and Mario Sanchez for helpful discussions.
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Appendix A: Technical Estimates
Appendix A: Technical Estimates
The following approximations are used in the numerical estimates above.
Lemma 11
For all \(n\ge 10\), we have
and
Proof
The central binomial coefficients satisfy
Therefore, by (3), we have
for all \(n\ge 10\). Similarly,
for all \(n\ge 10\), and (16) follows.
Next, observe that, for \(n\ge 10\), we have by (16) that
and
and so (17) follows. \(\square \)
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Kolesnik, B. The Asymptotic Number of Score Sequences. Combinatorica 43, 827–844 (2023). https://doi.org/10.1007/s00493023000374
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DOI: https://doi.org/10.1007/s00493023000374
Keywords
 Asymptotic enumeration
 Digraph
 Infinitely divisible distribution
 Lattice points
 Permutahedron
 Renewal sequence
 Reverse renewal theorem
 Score sequence
 Tournament