Monatsh Math DOI 10.1007/s00605-017-1128-9

Orbit Dirichlet series and multiset permutations Angela Carnevale1 · Christopher Voll1

Received: 4 January 2017 / Accepted: 25 October 2017 © Springer-Verlag GmbH Austria 2017

Abstract We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz’s q-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series’ abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture. Keywords Orbit Dirichlet series · Multiset permutations · Carlitz’s q-Eulerian polynomials · Hadamard products of rational generating functions · Igusa functions · local functional equations · Natural boundaries Mathematics Subject Classification 37C30 · 37P35 · 30B50 · 11M41 · 05A15 · 05A19

Communicated by A. Constantin.

B

Christopher Voll [email protected] Angela Carnevale [email protected]

1

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

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A. Carnevale, C. Voll

1 Introduction and main results Let X be a space and T : X → X a map. A closed orbit of length n ∈ N is a set of the form {x, T (x), T 2 (x), . . . , T n (x) = x} of cardinality n. Assume that the number OT (n) of closed orbits of length n under T is finite for all n ∈ N. The orbit Dirichlet series of T is the Dirichlet generating series dT (s) =

∞ 

OT (n)n −s ,

n=1

where s is a complex variable. If T has a single orbit of each length n, then dT (s) is just Riemann’s zeta  closed −s . If, more generally, T = T is such that the number of n function ζ (s) = ∞ r n=1 closed orbits of length n equals the number an (Zr ) of subgroups of Zr of index n for all n ∈ N, then dTr (s) is the well known Dirichlet generating series (or “zeta function”) ζZr (s) enumerating subgroups of finite index of the free abelian group Zr of rank r . More precisely, dTr (s) = ζ (s) =

∞ 

Zr

an (Z )n r

−s

=

n=1

r −1

ζ (s − i);

(1.1)

i=0

cf. [12, Proposition 1.1]. m mLet λ = (λ1 , . . . , λm ) ∈ N with λ1 ≥ · · · ≥ λm ≥ 1 be a partition of N = i=1 λi . For i = 1, . . . , m, let Tλi be a map as above with dTλi (s) = ζZλi (s). We write Tλ = Tλ1 × · · · × Tλm for the Cartesian product of the maps Tλi . Clearly, the arithmetic function n → OTλ (n) is multiplicative, whence  dTλ (s) = dTλ , p (s), p prime

where, for a prime p, dTλ , p (s) =

∞ 

OTλ ( p k ) p −ks .

k=0

We remark that maps Tλi as above exist, even if they are required to be smooth. Indeed, by a result of Windsor, any sequence (an )n≥1 of nonnegative integers may be realized as the sequence (OT (n))n≥1 for a suitable C ∞ -diffeomorphism T of the 2-dimensional torus X = T2 = (R/Z)2 ; cf. [26].

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Orbit Dirichlet series and multiset permutations

In this paper we prove and exploit combinatorial formulae for the Euler factors of orbit Dirichlet series of the form dTλ (s) above using generating polynomials for statistics on multiset permutations. Our first main result is phrased in terms of the bivariate polynomial Cλ ∈ Z[x, q] giving the joint distribution of the statistics des and maj on Sλ , the set of multiset permutations of the multiset {1, . . . 1, 2, . . . 2 . . . , m, . . . , m}. See Sect. 2 for precise          λ1

λ2

λm

definitions. Theorem 1.1 Let λ = (λ1 , . . . , λm ) be a partition of N . Then dTλ (s) =



Cλ ( p −1−s , p)

N

p prime

i=1 (1 −

pi−1−s )

=





w∈Sλ

p (−1−s) des(w)+maj(w)

N

i=1 (1 −

p prime

pi−1−s )

. (1.2)

Key to Theorem 1.1 is an identity, essentially due to MacMahon, for Hadamard products of certain rational generating functions. It is well known that if A(x) = ∞ ∞ k and B(x) = k ∈ Q(x) are rational functions, then their a x kx k=0 k k=0 b k Hadamard product (A ∗ B)(x) = ∞ k=0 ak bk x is also a rational function; cf. [23, m A (x) Proposition 4.2.5]. Given rational functions A1 (x), . . . , Am (x), we write ∗i=1 i for their Hadamard product A1 (x) ∗ · · · ∗ Am (x). Proposition 1.2 (MacMahon; cf. Remark 3.2) Let λ = (λ1 , . . . , λm ) be a partition of N . Then m ∗i=1

λi  k=0

Cλ (x, q) 1 = N ∈ Q(x, q). k i 1−q x i=0 (1 − xq )

We call a partition of the form λ = (r, . . . , r ) = (r m ) a rectangle. We use Theorem 1.1 to prove that the Euler factors in (1.2) satisfy certain functional equations upon inversion of the prime if and only if the partition λ is a rectangle. We denote by Tr×m the m-fold Cartesian power Tr × · · · × Tr . Theorem 1.3 Let p be a prime. For all r, m ∈ N, r +1 2

dTr×m , p (s)| p→ p−1 = (−1)r m p m (

)−r −r s d

Tr×m , p (s).

(1.3)

If λ is not a rectangle, then dTλ , p (s) does not satisfy a functional equation of the form dTλ , p (s)| p→ p−1 = ± p d1 −d2 s dTλ , p (s)

(1.4)

for d1 , d2 ∈ N0 . We prove Theorems 1.1 and 1.3 in Sect. 3. The functional equations (1.3) are deduced from the combinatorial properties of the polynomials Cλ studied in Sect. 2.

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In Sect. 4 we collect a number of corollaries about the analytic properties of the orbit Dirichlet series that we study. In particular, we determine the abscissa of convergence of dTλ (s) and establish meromorphic continuation beyond this abscissa. A standard application of a Tauberian theorem then yields an asymptotic result on the growth of the (partial sums of the) numbers OTλ (n); see Theorem 4.1. If λ = (r ) or λ = (1, 1), then dTλ (s) has meromorphic continuation to the whole complex plane. In contrast, for partitions with more than two parts—pertaining to Cartesian products of more than two maps—or two parts of equal length  than

mgreater λ − 2. 1 we establish a natural boundary for meromorphic continuation at i i=1 For partitions with two parts of unequal lengths, we establish such a natural boundary subject to a combinatorial conjecture on some special values of the polynomials Cλ discussed in Sect. 2.2; see Theorem 4.2. In Sect. 5 we concentrate on partitions of the form λ = (1m ), pertaining to the mth Cartesian power of a map with orbit Dirichlet series dT1 (s) = ζ (s). Orbit Dirichlet series of products of such maps were previously studied, for very special cases, in [17]. The result [17, Theorem 4.1], for instance, is the special case λ = (1, 1, 1) of our Theorem 4.2; see also Sect. 5. For partitions of the form λ = (1m ) the polynomial Cλ is the well-studied Carlitz’s q-Eulerian polynomial, enumerating the elements of the symmetric group by the statistics des and maj. We also observe that in this case the Euler factors of (1.2) are Igusa functions in the terminology of [19]. In Sect. 6 we consider “reduced” orbit Dirichlet series and note some connections with the theory of h-vectors of simplicial complexes. Dirichlet generating series are widely used in enumerative problems arising in algebra, geometry, and number theory. Orbit Dirichlet series as defined above are studied for instance in [7]. Local functional equations such as the ones established in Theorem 1.3 occur frequently in the theory of zeta functions of groups, rings, and modules; see, for example, [24,25]. In the cases where they are explained combinatorially, they may often be traced back to functional equations satisfied by Igusa-type functions; see, for instance, [14,19]. 1.1 Notation We write N = {1, 2, . . .} and, for a subset I ⊆ N, set I0 = I ∪ {0}. Given n ∈ N, we write [n] = {1, . . . , n} and n − I = {n − i | i ∈ I }. For I = {i 1 , . . . , ir } ⊆ [n − 1] with i 1 ≤ · · · ≤ ir we let n! n = I i 1 !(i 2 − i 1 )! · · · (n − ir )! denote the multinomial coefficient. Given a ≥ b ∈ N0 and a variable q, we write b  a q a−b+i − 1 ∈ Z[q] = b q qi − 1 i=1

for the q-binomial coefficient.

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2 Permutations of multisets In this section we set up notation and prove some basic facts regarding multiset permutations (see also [15, Section 5.1.2]). 2.1 Multiset permutations Let λ = (λ1 , . . . , λm ) be a partition of N =

m

i=1 λi .

The multiset

Aλ = {1, . . . 1, 2, . . . 2 . . . , m, . . . , m}          λ1

λ2

λm

comprises λ1 (indistinguishable) copies of the “letter” 1, λ2 copies of the “letter” 2 etc. A multiset permutation (or multipermutation) on Aλ is a word w = w1 . . . w N formed with all the N elements of Aλ . We denote by Sλ the set of all multiset permutations on Aλ . If λ = (1, . . . , 1) = (1m ), then we recover the set Sm of permutations of the set A(1m ) = {1, 2, . . . , m}. In general, Sλ lacks a natural group structure, but a number of classical statistics on the Coxeter group Sm have analogues N for general partitions. For instance, one defines wi ∈ Sλ as the descent set Des(w) of w = i=1 Des(w) = {i ∈ [N − 1] | wi > wi+1 }, where, of course, one uses the “natural” ordering m > · · · > 2 > 1 on the letters of A. The descent and major index statistics on Sλ are defined via des(w) = | Des(w)|

and

maj(w) =



i.

i∈Des(w)

The “trivial word” 1λ1 2λ2 . . . mλm is clearly the unique element in Sλ with empty descent set. Example 2.1 For λ = (3, 3, 1), the element w = 1212312 ∈ Sλ has Des(w) = {2, 5}, whence des(w) = 2 and maj(w) = 7. Remark 2.2 One may, more generally, consider multisets indexed by compositions, rather than partitions, of N . As we are interested in the joint distribution of des and maj, the order of the parts does not matter to us (cf. (2.3) below), so we only consider partitions. Recall that we call λ a rectangle if λ1 = · · · = λm = r , say, viz. λ = (r m ). In this case, we write Sr,m for S(r m ) . If, moreover, r = 1, then we write Sm for S1,m = S(1m ) , the (set underlying the) symmetric group of degree m. Lemma 2.3 The partition λ is a rectangle if and only if there exists a unique element of Sλ at which des attains its maximum. If λ = (r m ), then both des and maj take

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their maximal values at w0 = (m . . . 21)r of Sλ , viz. des(w0 ) = r (m − 1) and m 2 maj(w0 ) = r 2 . Proof Set s = λ1 and write μ = (μ1 , . . . , μs ) for the dual partition  of λ. Thus m = μ1 ≥ · · · ≥ μs ≥ 1. The statistic des(w) attains its maximal value sσ =1 (μσ −1) precisely at the word w0 = (μ1 . . . 21)(μ2 . . . 21) . . . (μs . . . 21) and all the elements of Sλ obtained from w0 by permuting the s “blocks” μσ . . . 21, m ). In this σ ∈ [s]. All these elements coincide if and only if λ is a rectangle, say λ = (r

 r m r case, μ = (m ) and w0 satisfies des(w0 ) = r (m − 1) and maj(w0 ) = 2 − m r2 =

 r 2 m2 .  We define the involution ◦

: Sr,m → Sr,m , w = w1 . . . w N → w ◦ = (m + 1 − w N ) . . . (m + 1 − w1 ) (2.1)

which “reverses and inverts” the elements of Sr,m . Remark 2.4 If r = 1, then w0 ∈ Sm is the “longest element” (with respect to Coxeter length) and ◦ is just conjugation by w0 . We collect some properties of this involution in the following elementary and easy lemma, whose proof we omit. Lemma 2.5 For all w ∈ Sr,m the following hold. (1) Des(w ◦ ) = r m − Des(w), (2) des(w ◦ ) = des(w), (3) maj(w ◦ ) = des(w)r m − maj(w). 2.2 Generating polynomials Let x and q be variables and set Cλ (x, q) =



x des(w) q maj(w) ∈ Z[x, q].

(2.2)

w∈Sλ

A result of MacMahon ([16, §462, Vol. 2, Ch. IV, Sect. IX]) states that, in Q(x, q) ∩ Q(q)x, m  ∞   λi + k Cλ (x, q) . (2.3) x k = N i k q i=0 (1 − xq ) k=0

(r m )

i=1

is a rectangle, then we write Cr,m for C(r m ) . If, moreover, r = 1, If λ = then we write Cm for C1,m = C(1m ) . In this case, (2.2) defines Carlitz’s q-Eulerian polynomial ([1,2])

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Orbit Dirichlet series and multiset permutations

Cm (x, q) =



x des(w) q maj(w) ∈ Z[x, q].

w∈Sm

Note that Cm (x, 1) =



x des(w) = Am (x)/x ∈ Z[x],

(2.4)

w∈Sm

where Am (x) is the mth Eulerian polynomial; cf. [23, Section 1.4]. Example 2.6 For λ = (2, 1), Sλ = {112, 121, 211}, so C(2,1) (x, q) = 1 + xq + xq 2 . For r = m = 2, S2,2 = {1122, 1221, 1212, 2112, 2211, 2121}, whence C2,2 (x, q) = 1 + xq + 2xq 2 + xq 3 + x 2 q 4 . Finally, for m = 3 resp. m = 4, C3 (x, q) = 1 + 2xq + 2xq 2 + x 2 q 3 , resp. C4 (x, q) = (1 + xq 2 )(1 + 3xq + 4xq 2 + 3xq 3 + x 2 q 4 ). To establish some of the analytic properties of dTλ (s) in Sect. 4, we need a description of the unitary factors of the bivariate polynomials Cλ (x, q). Here, a polynomial f ∈ Z[x, q] is called unitary if it is nonconstant and there exists F ∈ Z[Y ] such that all complex roots of F have absolute value 1 and f (x, q) = F(x a q b ) for some a, b ∈ N0 . As maj(w) > 0 implies des(w) > 0 for all w ∈ Sλ , unitary factors of Cλ (x, q) ∈ Z[x, q] give rise to unitary factors of Cλ (x, 1) =



x des(w) ∈ Z[x].

(2.5)

w∈Sλ

The following Lemma describes the occurrence of unitary factors of Carlitz’s qEulerian polynomials, pertaining to partitions of the form λ = (1m ). Lemma 2.7 Carlitz’s q-Eulerian polynomial Cm (x, q) ∈ Z[x, q] has a unitary factor if and only if m is even. If m = 2k, then Cm (x, q) = (1 + xq k )Cm (x, q),  {k} {k} where Cm (x, q) = w∈S {k} x des(w) q maj(w) and Sm is the parabolic quotient Sm = m {w ∈ Sm | Des(w) ⊆ [m − 1]\{k}}. Moreover, Cm (x, q) has no unitary factor. Proof By a result of Frobenius ([9, p. 829]), the roots of Cm (x, 1) are all real, simple, and negative; moreover, −1 is a root only for even m. Thus the q-Eulerian polynomials Cm (x, q) have unitary factors only for even m. Let m = 2k and denote by w0 the

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A. Carnevale, C. Voll {k}

{k}

longest element of Sm . The map Sm → Sm \Sm , w → ww0 , is obviously a bijection. Hence    Cm (x, q) = x des(w) q maj(w) = xq j w∈Sm

=





w∈Sm j∈Des(w)

{k} w∈Sm j∈Des(w)

=







xq + j







xq j + xq k



xq j

{k} w∈Sm \Sm j∈Des(w) j=k

xq j + xq k

{k} w∈Sm j∈Des(w)

= (1 + xq k )

xq j

{k} w∈Sm \Sm j∈Des(w)

{k} w∈Sm j∈Des(w)

=









xq j

{k} w∈Sm j∈Des(w)



xq j = (1 + xq k )Cm (x, q).

{k} w∈Sm j∈Des(w)

The non-existence of unitary factors of Cm (x, q) follows again from the simplicity of  x = −1 as a root of Cm (x, 1). Remark 2.8 The polynomials Cm (x, 1), for m odd, resp. Cm (x, 1)/(1 + x), for m even, have been conjectured to be irreducible for all m; for a discussion and proofs of irreducibility in various special cases, see [13]. Remark 2.9 Consider again a general partition λ. Generalizing the result of Frobenius referred to in the proof of Lemma 2.7, all zeros of the polynomials Cλ (x, 1) are real, simple, and negative; see [20, Corollary 2]. By the above discussion, a necessary condition for the occurrence of unitary factors of Cλ (x, q) is hence that Cλ (−1, 1) = 0. We remark that in the case that λ = (r m ) is a rectangle, Cλ (−1, 1) is, up to a sign, the so-called Charney-Davis quantity of the graded poset of the disjoint union of m labelled chains of length r ; see [18]. For our applications in Sect. 4 we require statements about the (non-)existence of unitary factors of Cλ (x, q) principally in the case m = 2, on which we focus for most of the reminder of this section. Recall that C(λ1 ,λ2 ) (x, 1) is the descent polynomial of S(λ1 ,λ2 ) ; cf. (2.5). In [16, §144–146, Vol. 1, Ch. II, Sect. IV] MacMahon gives three proofs of the following lemma. Lemma 2.10 (MacMahon) Let λ = (λ1 , λ2 ). Then C(λ1 ,λ2 ) (x, 1) =

λ2  λ1 λ2 j=0

j

j

x j.

In terms of Jacobi polynomials, (0,λ1 −λ2 )

C(λ1 ,λ2 ) (x, 1) = (1 − x)λ2 Pλ2

123



1+x 1−x

;

Orbit Dirichlet series and multiset permutations

cf. [10, eq. (1.2.7)]. It follows from MacMahon’s third proof of Lemma 2.10 (cf. [16, §146]) that the number of elements in S(λ1 ,λ2 ) with k descents equals the number of elements with k occurrences of 2 in the first λ1 positions. We conjecture the following. Conjecture A Let λ1 > λ2 . Then the following equivalent statements hold: (1) C(λ1 ,λ2 ) (−1, 1) = 0, (0,λ −λ ) (2) Pλ2 1 2 (0) = 0, (3) |{w ∈ S(λ1 ,λ2 ) with an even number of 2s in the first λ1 positions}| = |{w ∈ S(λ1 ,λ2 ) with an odd number of 2s in the first λ1 positions}|. In particular, C(λ1 ,λ2 ) (x, q) has no unitary factor. D. Stanton pointed out to us that C(λ1 ,λ2 ) (−1, 1) = 0 for all (λ1 , λ2 ) satisfying λ1 > λ2 (λ2 + 1) − 1, since the alternating summands have increasing absolute values; L. Habsieger informed us of unpublished work of his establishing Conjecture A for all λ1 ≥ cλ2 for some constant c > 1. The quantity C(λ1 ,λ2 ) (−1, 1) may also be expressed in terms of Krawtchouk polynomials; it is equal to (−1)λ2 kλ2 (λ2 , 2, λ1 + λ2 ) in the notation of [4]. If λ1 = λ2 , then C(λ1 ,λ2 ) (−1, 1) = 0 if and only if the λi are even (see [18, Proposition 2.4]), whence C(λ1 ,λ2 ) (x, q) has no unitary factors in this case. In the odd case, the following holds. Proposition 2.11 ([3, Proposition 5]) Let λ1 = λ2 = r ≡ 1 (mod 2). Then  (x, q), Cr,2 (x, q) = (1 + xq r )Cr,2  (x, q) has no unitary factor. where Cr,2

Generalizing Lemma 2.7, Conjecture A and Proposition 2.11, we put forward the following Conjecture B Let λ = (λ1 , . . . , λm ) be a partition. Then Cλ (x, q) has a unitary factor if and only if λ = (r m ) is a rectangle, with r odd and m even. In this case, Cr,m (x, q) = (1 + xq

rm 2

 )Cr,m (x, q)

 (x, q) has no unitary factor. and Cr,m

2.3 Functional equations Proposition 2.12 For all r, m ∈ N, 2 m Cr,m (x −1 , q −1 ) = x −r (m−1) q −r ( 2 ) Cr,m (x, q).

If λ is not a rectangle, then Cλ (x, q) does not satisfy a functional equation of the form

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A. Carnevale, C. Voll

Cλ (x −1 , q −1 ) = x −d1 q −d2 Cλ (x, q),

(2.6)

for d1 , d2 ∈ N0 . Proof As Cλ (x, 1) ∈ Z[x] has constant term 1, a necessary condition for Cλ to satisfy a functional equation of the form (2.6) is that Cλ (x, 1) is monic. By Lemma 2.3, this holds if and only if λ is a rectangle. This proves the second statement. To establish the first statement, let r, m ∈ N. For i ∈ [r (m − 1)]0 , we set 

(i) (q) = Cr,m

q maj(w) ∈ Z[q],

{w∈Sr,m |des(w)=i}

r (m−1)

so that Cr,m (x, q) = yields

i=0



(i) Cr,m (q −1 ) = q −ir m

(i) Cr,m (q)x i . With the map ◦ defined in (2.1), Lemma 2.5



q ir m−maj(w) = q −ir m

{w∈Sr,m | des(w)=i}

◦

(i) q maj(w ) = q −ir m Cr,m (q).

{w∈Sr,m | des(w)=i}

Using this and the relations 2 m (r (m−1)−i) (i) (q) = q r ( 2 )−ir m Cr,m (q) Cr,m

(cf. [16, §461, Vol. 2, Ch. IV, Sect. IX]) we obtain Cr,m (x

−1

,q

−1

)=

r (m−1) 

(i) Cr,m (q −1 )x −i

i=0

=

r (m−1) 

=

r (m−1) 

(i) q −ir m Cr,m (q)x −i

i=0 2 m (r (m−1)−i) q −ir m q −r ( 2 )+ir m Cr,m (q)x −i

i=0 2 m = q −r ( 2 )

r (m−1) 

( j)

Cr,m (q)x −r (m−1)+ j

j=0 2 m = x −r (m−1) q −r ( 2 ) Cr,m (x, q).



3 Proofs of Theorems 1.1 and 1.3 3.1 Proof of Theorem 1.1 Let r ∈ N. Recall that ζZr (s) = p prime ζZrp (s), where, for a prime p, the Euler factor  r −ks enumerates the Z -submodules of finite additive index ζZrp (s) = ∞ p k=0 a p k (Z p ) p r in Z p . Here, Z p denotes the ring of p-adic integers.

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Orbit Dirichlet series and multiset permutations

Lemma 3.1 (cf., e.g., [11]) For all k ∈ N0 ,

r −1+k . k p

a pk (Zrp ) = Proof For a variable t, ∞ 

a pk (Zrp )t k

k=0

∞  r −1+k 1 = = r tk; i−1 t) k p i=1 (1 − p

(3.1)

k=0

see (1.1) for the first equality and, for instance, [23, Ch. 1.8] for the second.



Remark 3.2 Proposition 1.2 follows from combining the second equality in (3.1) with (2.3). Recall that for a map T : X → X we denote by OT (n) the number of closed orbits of T of length n. Let FT (n) = |{x ∈ X | T n (x) = x}| denote the number of points of period n. Then  dOT (d) (3.2) FT (n) = d|n

and, by Möbius inversion, OT (n) =

1  n  FT (d). μ n d

(3.3)

d|n

From (3.2), pT (s) :=

∞ 

FT (n)n −s = ζ (s)dT (s − 1).

(3.4)

n=1

Let now r ∈ N and Tr be a map with orbit Dirichlet series dTr (s) = ζZr (s) as in (1.1). Corollary 3.3 For all k ∈ N0 , FTr ( p k ) =

k 

p j a p j (Zrp ) =

j=0

Proof By (1.1) and (3.4), pTr (s) = ζ (s) Lemma 3.1, this yields the result.

r −1 i=0



r +k . k p

ζ (s + 1 − i) = ζZr +1 (s). Together with 

Recall that λ = (λ1 , . . . , λm ) is a partition. For all n ∈ N we have FTλ1 ×···×Tλm (n) = m i=1 FTλi (n), so that, by Corollary 3.3, for a prime p and k ∈ N0 , FTλ ( p k ) =

m  λi + k . k p i=1

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A. Carnevale, C. Voll

Using (3.3) we deduce, as in the proof of [17, Proposition 3.1], that, for k > 0, k 1  p FTλ (d) OTλ ( p ) = k μ p d k k

d| p

 1  = k FTλ ( p k ) − FTλ ( p k−1 ) p m

 m  λi + k − 1 1  λi + k − . = k p k k−1 p p i=1

i=1

Hence m

 ∞ m   λi + k  λ 1 + k − 1 i dTλ , p (s) = OTλ ( p k )t k = 1 + − tk pk k k−1 p p k=0 k=1 i=1 i=1 m



 k ∞  t λi + k t = 1− . p p k p ∞ 

k=0

i=1

By substituting (t/ p, p) for (x, q) in (2.3) and setting t = p −s , this may be rewritten as Cλ ( p −1−s , p) . dTλ , p (s) = N i−1−s ) i=1 (1 − p The second statement in (1.2) follows from (2.2). This concludes the proof of Theorem 1.1. 3.2 Proof of Theorem 1.3 Given the expression (1.2) in Theorem 1.1, it is clear that a functional equation of the form (1.4) holds if and only if Cλ (x, q) satisfies a functional equation of the form (2.6). By Proposition 2.12, this holds if and only if λ is a rectangle. If λ = (r m ), then, substituting ( p −1−s , p) for (x, q), this result implies that 2 m Cr,m ( p 1+s , p −1 ) = p −r ( 2 )+r (m−1)+sr (m−1) Cr,m ( p −1−s , p).

(3.5)

The functional equation (1.3) holds, as r m+1 1 1 = (−1)r m p ( 2 )−r m−sr m r m −i+1+s (1 − p ) (1 − pi−1−s ) i=1 i=1

r m

combined with (3.5) gives dTr×m , p (s)| p→ p−1 = (−1)r m p (

r m+1 2

123

)−r m−r 2 (m2 )+r (m−1)−sr m+sr (m−1) d

Tr×m , p (s)

Orbit Dirichlet series and multiset permutations r +1 2

= (−1)r m p m (

)−r −r s d

Tr×m , p (s).

4 Analytic properties and asymptotics In this section we exploit the combinatorial description of the Dirichlet series dTλ (s) given in  (1.2) to deduce some of their key analytic properties. Recall that λ is a partition m λi . In the following, f (n) ∼ g(n) means that limn→∞ f (n)/g(n) = 1. of N = i=1 Theorem 4.1 (1) The orbit Dirichlet series dTλ (s) has abscissa of convergence N . If m = 1 or λ = (1, 1), then it may be continued meromorphically to the whole complex plane; otherwise it has meromorphic continuation to {s ∈ C | Re(s) > N − 2}. In any case, the continued function is holomorphic on the line {s ∈ C | Re(s) = N } except for a simple pole at s = N . (2) There exists a constant K λ ∈ R>0 such that  ν≤n

OTλ (ν) ∼ K λ n N

as n → ∞.

Proof Recall that, for all b ∈ N0 , the translate ζ (s − b) of Riemann’s zeta function converges for Re(s) > b + 1 and may be continued meromorphically to the whole complex plane. The continued function is holomorphic on the line {s ∈ C | Re(s) = b + 1} except for a simple pole at s = b + 1. This establishes all claims in (1) if m = 1 2 ζ (s−1) or λ = (1, 1), as dTr (s) = ζZr (s) and dT(1,1) (s) = ζ (s)ζ (2s) . Assume thus that m ≥ 2 and λ = (1, 1) and recall the expression (1.2) for dTλ (s). The product p prime Cλ ( p −1−s , p) has abscissa of convergence N − 1 and may be meromorphically continued to {s ∈ C | Re(s) > N − 2}. Indeed, an Euler product of the form ⎛ ⎞   ⎝1 + pi−ks ⎠ , (i,k)∈I

p prime

where I ⊂ N0 × N is a finite (multi-)set, converges on {s ∈ C | Re(s) > α}, where  α = max

 i +1 | (i, k) ∈ I , k

and has a meromorphic continuation to {s ∈ C | Re(s) > β}, where  β = max

 i | (i, k) ∈ I ; k

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A. Carnevale, C. Voll

see [6, Lemmas 5.4 and 5.5]. It follows from inspection of the Euler product 

Cλ ( p −1−s , p) =

p prime

⎛





⎝



⎞ p j−1−s ⎠

(4.1)

w∈Sλ j∈Des(w)

p prime

that the relevant maxima α = N − 1 resp. β = N − 2 are both attained at the elements w ∈ Sλ with Des(w) = {N − 1}. We note that |{w ∈ Sλ | Des(w) = {N − 1}}| = m − 1.

(4.2)

As  p prime

N

1

i=1 (1 −

pi−1−s )

=

N 

ζ (s − i + 1)

i=1

has abscissa of convergence N > α, this concludes the proof of (1). Statement (2) follows from (1), for instance via the Tauberian theorem [5, Theorem 4.20].  If m > 1 and λ = (1, 1), then the meromorphic continuation to β = N − 2 is often—and conjecturally always—best possible, as we now prove. Theorem 4.2 Assume that λ = (1, 1) and that either (i) m > 2 or (ii) m = 2 and λ1 = λ2 or (iii) m = 2, λ1 > λ2 , and Conjecture A holds. Then the orbit Dirichlet series dTλ (s) has a natural boundary at {s ∈ C | Re(s) = N − 2}. Proof We keep the notation as in Theorem 4.1 and set W λ (X, Y ) = Cλ (X −1 Y, X ) =



ci,k X i Y k ∈ Z[X, Y ]

(i,k)∈Iλ

for suitable Iλ ⊆ N20 and ci,k ∈ N. The Euler product (4.1) then reads 

W λ ( p, p −s ).

(4.3)

p prime

To prove that under any of the assumptions (i)–(iii), the Euler product (4.3) has a natural boundary at β = N − 2 we consider the ghost polynomial associated to W λ and prove that W λ is a polynomial of type I [in case (i)] or type II [in cases (ii) and (iii)] in the terminology of [6, Section 5.2].

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Orbit Dirichlet series and multiset permutations

We claim that the first factor of the ghost polynomial of W λ (X, Y ) is, in notation close to the one used in [6, Section 5.2], 

1λ (X, Y ) = W

ci,k X i Y k = 1 + (m − 1)X β Y.

(4.4)

(i,k)∈l1 ∩Iλ

Here, l1 is the line in R2 through (0, 0) and (β, 1). It is characterized by the fact that its gradient 1/β is minimal among the lines in R2 passing through (0, 0) and the points (i, k) ∈ Iλ \{(0, 0)}. Moreover, l1 ∩ Iλ = {(0, 0), (β, 1)} and cβ,1 = m − 1 (cf. (4.2)), which proves (4.4). Setting U = X β Y , we obtain 1λ (U ) = 1 + (m − 1)U ∈ Z[U ]. W  λ (U ) is not cyclotomic, whence W λ (X, Y ) is a polynomial of type If m > 2, then W 1 I in the parlance of [6, p. 127]. Without loss of generality we may divide W λ by any unitary factors it may have; cf. Conjecture B. Indeed, if W λ = f V λ for f ∈ Z[X, Y ] unitary, then the Newton polygon of W λ is the Minkowski sum of the Newton polygons  λ does not have of f and V λ . The former, however, is a segment of a line in R2 . As W 1 λ , λ = V a unitary factor, the slope of this line is strictly larger than 1/β, whence W 1 1 λ λ i.e. the first factors of the ghosts of W and V coincide. Assuming thus, as we may, that W λ has no unitary factors, [6, Theorem 5.6] yields that β is a natural boundary for (4.3) and thus for dTλ (s). This concludes the proof in case (i).  λ (U ) = 1 + U Turning to cases (ii) and (iii) we now assume that m = 2. Hence W 1 λ is cyclotomic. In particular, W (X, Y ) is not of type I. We claim that it is a polynomial of type II in the sense of [6, p. 127]. To prove this, we check that the hypotheses of [6, Corollary 5.15] are satisfied. To this end, we verify that W λ (X, Y ) is such that Hypotheses 1 and 2 defined on [6, p. 134] are satisfiable. The polynomial A(U ) =  1 + nk =β cn k ,k U k = 1 + U (cf. [6, p. 134]) obviously has a unique root ω = −1. k It is simple, so in particular satisfiesHypothesis  1. Bγ (ω) Hypothesis 2 is equivalent to Re − ω A (ω) < 0, hence to Bγ (ω) < 0, where γ := min{n ∈ N0 | Bn (ω) = 0} and, for n ∈ N0 and (n j , j) ∈ Iλ such that n j /j = β and j is minimal with this property,   ci,k U k = ci,k U k ; Bn (U ) = n j k−i j=n

βk−i=n

cf. [6, (5.12)]. Note that B0 (U ) = A(U ) = 1 + U . Recall that   W λ (X, Y ) = X maj(w)−des(w) Y des(w) = ci,k X i Y k . w∈Sλ

(i,k)∈Iλ

For (i, k) ∈ Iλ , we thus have ci,k = |{w ∈ Sλ | k = des(w), i = maj(w) − k}|.

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A. Carnevale, C. Voll

We claim that γ = 1. If (i, k) satisfies βk − i = 1,

(4.5)

then clearly k = 0. If k = 2, then (4.5) necessitates i = 2N − 5, that is maj(w) = 2N − 3. But there is no element w ∈ Sλ such that des(w) = 2 and maj(w) = 2N − 3. Indeed, such an element would need to have descents at the consecutive positions N − 2 and N − 1 (as maj(w) = 2N − 3 = (N − 1) + (N − 2)), which is clearly impossible for a word in {1, . . . , 1, 2, . . . , 2}. A similar argument excludes pairs (i, k)       λ1

λ2

that satisfy (4.5) and for which k > 2. To determine B1 (U ) we thus need to determine cβ−1,1 = |{w ∈ Sλ | des(w) = 1, maj(w) = N − 2}|, i.e. to enumerate the multiset permutations with no descent in the first N − 3 positions and ending in . . . 212 or . . . 211. If λ2 > 1, then there are exactly two such words; if λ2 = 1, then only the second option occurs. So B1 (U ) = 2U resp. B1 (U ) = U . In any case, γ = 1 and Bγ (ω) = −2 < 0 resp. Bγ (ω) = −1 < 0. Hence Hypothesis 2 is satisfied. Since Hypotheses 1 and 2 are satisfied and 1 = γ ≥ j = 1, [6, Corollary 5.15] implies that W λ (X, Y ) is of type II. Thus in case (ii) for λ1 = λ2 odd, and in case (iii), as W λ (X, Y ) has no unitary factors, [6, Theorem 5.6] yields that β is a natural boundary for (4.3) and thus for dTλ (s). In case (ii) for λ1 = λ2 even, Proposition 2.11 asserts that a unique unitary factor exists: W λ (X, Y ) = (1 + X λ1 −1 Y )W  λ (X, Y ) for some W  λ ∈ Z[X, Y ]. But β = N − 2 > λ1 − 1, so the minimal gradient for W  λ (X, Y ) is still β. Thus also in this case [6, Theorem 5.6] implies that β is a natural boundary. 

5 Connection with Igusa functions and the special case λ = (1m ) Taking λ = (1m ) corresponds to considering the mth power of a map T = T1 whose orbit Dirichlet series dT (s) is the Riemann zeta function ζ (s). In this case, Theorem 1.1 reads dT ×m (s) =

 p prime

 Cm ( p −1−s , p) m = i−1−s ) i=1 (1 − p



p prime

(−1−s) des(w)+maj(w) w∈Sm p m . i−1−s ) i=1 (1 − p

where Cm (x, q) = C1,m (x, q) is Carlitz’s q-Eulerian polynomial. More generally one may, for a ∈ R≥0 , consider a map a T such that da T (s) = ζ (s − a). Then Oa T ( p k ) = pak

and

Fa T ( p k ) =

k  j=0

123

p j pa j =



1+k . 1 pa+1

Orbit Dirichlet series and multiset permutations

The orbit Dirichlet series of the mth power of a T is thus MacMahon’s generating series (2.3) for λ = (1m ) and (x, q) = ( p −1−s , pa+1 ): 

Cm ( p −1−s , pa+1 ) m (a+1)i−1−s ) i=1 (1 − p p prime  (−1−s) des(w)+(a+1) maj(w)  w∈Sm p m = . (a+1)i−1−s ) i=1 (1 − p

da T ×m (s) =

(5.1)

p prime

A formula for the local factors da T ×m , p (s) of da T ×m (s) appears in [17, p. 41], where they are called E p (s), and suffers from a transcript error in the definition of the expression Ab . Moreover, no combinatorial interpretation is given. [17, Theorem 4.1] is Theorem 4.2 in the special case λ = (1, 1, 1). Each factor of the Euler product (5.1) is an instance of an Igusa function:  (a+1) j−1−s Cm ( p −1−s , pa+1 ) w∈Sm j∈Des(w) p = da T ×m , p (s) = m m (a+1)i−1−s ) (a+1)i−1−s ) i=1 (1 − p i=1 (1 − p

 m  p (a+1)i−1−s 1 = ∈ Q( p, p −s ). I 1 − p (a+1)m−1−s 1 − p (a+1)i−1−s I ⊆[m−1]

i∈I

m ) In the terminology of [19, Definition 2.5] it would be called Im (1; ( p (a+1)i−1−s )i=1 and so (1.3) in Theorem 1.3 follows in this case from [19, Proposition 4.2]. In the case of a general partition, we are not aware of a simple expression of the local factors of the orbit Dirichlet series of the product of such “shifted maps”. Turning back to the case a = 0 and general partition λ, the local factors of (1.2) may be rewritten as

dTλ , p (s) =

1 1 − p N −1−s



νλ,I

I ⊆[N −1]

 i∈I

pi−1−s ∈ Q( p, p −s ) 1 − pi−1−s

where νλ,I = |{w ∈ Sλ | Des(w) ⊆ I }|. We are not aware of a simple expression, say in terms of multinomial coefficients, for νλ,I if λ is not of the form (1m ).

6 Reduced orbit Dirichlet series: setting p = 1 Viewing the Euler factors of (1.2) as bivariate rational functions in p and t = p −s , one may evaluate them at p = 1 whilst leaving t as an independent variable. Motivated by the notion of reduced zeta functions of Lie algebras introduced in [8] we thus define the reduced orbit Dirichlet series dTλ ,red (t) :=

Cλ (t, 1) ∈ Q(t). (1 − t) N

It seems remarkable that for λ = (1m ) the reduced orbit Dirichlet series dT ×m ,red (t) 1 is the Hilbert series of the Stanley–Reisner ring of a simplicial complex. Indeed, let k

123

A. Carnevale, C. Voll

be any field, write sd( m−1 ) for the barycentric subdivision of the (m − 1)-simplex

m−1 —or, equivalently, the Coxeter complex of type Am−1 —, with Stanley–Reisner (or face) ring k[sd( m−1 )]; see, for instance, [22, Ch. III, Sec. 4]. The fact that the mth Eulerian polynomial (cf. (2.4)) is the generating function of the h-vector of sd( m−1 ) is reflected in the following fact. Proposition 6.1 dT ×m ,red (t) = 1

Am (t)/t = Hilb(k[sd( m−1 )], t). (1 − t)m

Similarly, we observe that for λ = (r, r ), the polynomial Cr,2 (t, 1) may be viewed as the h-vector of the r -dimensional type-B simplicial associahedron Q B n ; cf. [21, Corollary 1]. Acknowledgements We are grateful to Tom Ward, who pointed us to [17] and suggested to generalize the set-up there, and to Michael Baake, who introduced us to Ward. Both Baake and Ward made valuable comments about earlier drafts of this paper. We thank Laurent Habsieger, Christian Krattenthaler, Tobias Rossmann, and Dennis Stanton for helpful remarks. We were supported by the German-Israeli Foundation for Scientific Research and Development, through Grant No. 1246.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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