Restricted partition functions and identities for

Ramanujan J DOI 10.1007/s11139-017-9904-7

Restricted partition functions and identities for degrees of syzygies in numerical semigroups Leonid G. Fel1

Received: 12 November 2014 / Accepted: 4 March 2017 © Springer Science+Business Media New York 2017

Abstract We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series of numerical semigroup d1 , . . . , dm , m ≥ 2, generated by an arbitrary set of positive integers {d1 , . . . , dm }, gcd(d1 , . . . , dm ) = 1. These identities are obtained by studying the rational representation of the Hilbert series and the quasipolynomial representation of the Sylvester waves in the restricted partition function. In the cases of symmetric semigroups and complete intersections, these identities become more compact; for the latter we find a simple identity relating the degrees of syzygies with elements of generating set {d1 , . . . , dm } and give a new lower bound for the Frobenius number. Keywords Numerical semigroups · The Hilbert series · The Betti numbers Mathematics Subject Classification Primary 20M14 · Secondary 11P81

1 Introduction A study of the Diophantine equations is on the border-line between combinatorial number theory and commutative algebra. Most important results await at the intersection of the two theories, as was demonstrated in the last decades [23]. In this paper, we bring together two different approaches to the study of the linear Diophantine equations: the theory of restricted partitions and the theory of commutative semigroup rings. We show that such merging can be used to produce new results. It allows to establish a set of new quasipolynomial identities for degrees of syzygies

B 1

Leonid G. Fel [email protected] Department of Civil Engineering, Technion, Haifa 3200, Israel

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in the Hilbert series H (dm ; z) of numerical semigroup S (dm ) = d1 , . . . , dm  of any embedding dimension, m ≥ 3, generated by an arbitrary set of positive integers dm = {d1 , . . . , dm }, where gcd(d1 , . . . , dm ) = 1. The paper is organized in six sections and two appendices. In Sect. 2, we recall the basic facts about numerical semigroup S (dm ) and their Hilbert series H (dm ; z). For degrees of syzygies, we state the main result on polynomial identities (Theorem 1) which are independent of the structure of the generating set dm . Another result on quasipolynomial identities (Theorem 2) is valid when among the generators di of the set dm there exists a subset q (dm ) ⊂ dm such that q (dm ) := {di | q | di } and #q (dm ) ≥ 2. In Sect. 3, we recall the basic facts about the partition of a nonnegative integer s into positive integers {d1 , . . . , dm }, each not greater than s, and about the number of nonequivalent partitions, or representations (Reps) W (s, dm ). The main emphasis is done on the Sylvester waves Wq (s, dm ) and their symbolic Reps in [26]. We derive the quasipolynomial Reps of the Sylvester waves with trigonometric functions as coefficients, and prove that their leading terms cannot vanish simultaneously. (Proposition 1). In Sect. 4, we derive the relationship between the rational Rep of the Hilbert series H (dm ; z) and the quasipolynomial Rep of the Sylvester waves in the restricted partition function W (s, dm ). This allows to prove Theorem 1 (in Sect. 4.1) and Theorem 2 (in Sect. 4.2) on polynomial and quasipolynomial identities for degrees of syzygies. Then we prove Theorem 3, which relates the degrees of syzygies of a numerical semigroup and the values of a real periodic function. In Sect. 5, we discuss different applications of Theorems 1 and 2 to various kinds of numerical semigroups: complete intersections (CI), symmetric semigroups, and symmetric semigroups (not CI) generated by 4 and 5 elements.

  2 Numerical semigroups and Hilbert series H dm ; z Throughout the article we assume that the numerical semigroup S (dm ) is generated by a minimal set dm of positive integers with finite complement in N, # {N \ S (dm )} < ∞. We study the generating function H (dm ; z) of such semigroup S (dm ),   H dm ; z =

 s∈

zs ,

(2.1)

S(dm )

which is referred to as the Hilbert series of S (dm ). Recall the main definitions and facts on numerical semigroups which are necessary here. A semigroup 

S d

m



 = s ∈ N ∪ {0} | s =

m 

 xi di , xi ∈ N ∪ {0} , gcd(d1 , . . . , dm ) = 1,

i=1

is said to be generated by a minimal set of m natural numbers d1 < · · · < dm , if neither of its elements is linearly representable by the rest of the elements. It is classically known that d1 ≥ m [17], where d1 and m are called the multiplicity and the embedding

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dimension (edim) of the semigroup, respectively. If d1 = m, then S (dm ) is called a semigroup of maximal edim (MED). The Frobenius number of S (dm ) is defined by F (dm ) := min {s ∈ S (dm ) | s + N ∪ {0} ⊂ S (dm )} − 1. A semigroup S (dm ) is called symmetric if for any integer s the following con/ S (dm ). Otherwise S (dm ) is called dition holds: if s ∈ S (dm ) then F (dm ) − s ∈ nonsymmetric. Denote by G (dm ) the complement of S (dm ) in N, i.e., G (dm ) = N \ S (dm ), and call it the set of gaps. The cardinality (#) of G (dm ) is called the genus of S (dm ), g (dm ) := #G (dm ). For the set G (dm ), introduce the generating function  (dm ; z), which is related to the Hilbert series, 

   dm ; z =

    z s ,  dm ; z + H dm ; z =

s ∈ G(dm )

1 . 1−z

The Hilbert series H (dm ; z) of a numerical semigroup S (dm ) is a rational function [28]   Q (dm ; z) , H dm ; z = m  di i=1 1 − z

(2.2)

where H (dm ; z) has a pole z = 1 of order 1. The numerator Q (dm ; z) is a polynomial in z,         Q dm ; z = 1 − Q 1 dm ; z + Q 2 dm ; z − · · · + (−1)m−1 Q m−1 dm ; z , (2.3) β i (d )     Q i dm ; z = z C j,i , 1 ≤ i ≤ m − 1, deg Q i dm ; z < deg Q i+1 (dm ; z), m

j=1

(2.4) where the positive numbers C j,i and βi (dm ) denote the degrees of the ith syzygy and the ith Betti number, respectively. The latter satisfy the equality [28]           β0 dm − β1 dm + β2 dm − · · · − (−1)m βm−1 dm = 0 , β0 dm = 1. (2.5) Summands z C j,i in (2.4) stand for syzygies of different kinds. The degrees C j,i of homogeneous basic invariants for the syzygies of the ith kind are representable by elements of tuple dm , C j,i =

m 

h kj,i dk , h kj,i ∈ {N ∪ 0} ,

k=1

C j+1,i ≥ C j,i , Cβi+1 (dm ),i+1 > Cβi (dm ),i , C1,i+1 > C1,i ,   C j,i = Cr,i+2k−1 , 1 ≤ j ≤ βi dm ,

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  1 ≤ r ≤ βi+2k−1 dm , 1 ≤ k ≤



m −i . 2

(2.6)

An inequality in (2.6) means that all necessary cancellations (annihilations) of terms z C j,i in (2.3) are already performed. However, the other equalities, C j,i = Cr,i+2k and C j,i = Cq,i , j = q, are allowed for all syzygies, except the syzygy of the last (m − 1)st kind ([2], Lemma 2). A degree of the polynomial Q (dm ; z) is strongly related [2] to the Frobenius number by     deg Q dm ; z = F dm + σ1 , where σ1 = d1 + · · · + dm .

(2.7)

For rigorous definitions of syzygies of the first and higher kinds, their moduli and specifying homomorphisms, and their Betti numbers and minimal-free resolution, we refer to [23]. Regarding the degrees of syzygies in general case of nonsymmetric semigroups, their values C j,i are usually obtained by computational algorithms (e.g., [6]) calculating the homogeneous bases for the syzygies moduli and their specifying homomorphisms in a minimal-free resolution. If a semigroup S (dm ) is symmetric, then the duality relation for numerator Q (dm ; z) holds [2] 

  m (2.8) Q dm ; z −1 z deg Q(d ;z) = (−1)m−1 Q dm ; z , and by consequence of (2.8), we have       βk dm = βm−k−1 dm , C j,k + C j,m−k−1 = deg Q dm ; z ,

0 ≤ k < m, 1 ≤ j ≤ βk (dm ). (2.9)

In fact, the second equality in (2.9) does not contribute much to the determination of the degrees of syzygies, since in accordance with (2.7) the degree of Q (dm ; z) is related to the Frobenius number F (dm ) which cannot be expressed for m ≥ 3 in terms of generators di as an algebraic function. If β1 (dm ) = m − 1, then the semigroup S (dm ) is called complete intersection (CI) and the numerator of the Hilbert series is given by [28]        Q dm ; z = 1 − z e1 1 − z e2 · · · 1 − z em−1 , e j ∈ N .

(2.10)

There are a few types of semigroups where the Betti numbers βi (dm ) and degrees of completely. This is true, e.g., for MED semigroups [11,27] and syzygies C j,i are   known semigroups S d3 of a special kind: Pythagorean semigroups [9], pseudosymmetric semigroups [11], and Fibonacci and Lucas symmetric semigroups [10]. A family of semigroups with known Betti numbers but unknown degrees of syzygies is much  wider, e.g., nonsymmetric semigroups S d3 [16], symmetric semigroups S d4 [3], and symmetric semigroups of almost maximal edim, d1 = m + 1 ([27], Theorem 2) and CI [28].

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2.1 Main results We present two theorems on polynomial and quasipolynomial identities for the degrees C j,i of syzygies. Their proof will follow later, Theorem 1 in Sect. 4.1 and Theorem 2 in Sect. 4.2. Start with a generic case of a generating set dm , not assuming any relationships among d j . Theorem 1 Let the numerical semigroup S (dm ) be given with its Hilbert series H (dm ; z) in accordance with (2.2, 2.3). Then the following polynomial identities hold, β1 (dm )

C rj,1

−

β2 (dm )

j=1

C rj,2 + · · ·

j=1 βm−1 (dm )



+ (−1)m

C rj,m−1 = 0, r = 1, . . . , m − 2,

(2.11)

j=1 β1 (dm )

C m−1 j,1

−

j=1

β2 (dm )

C m−1 j,2 + · · ·

j=1 βm−1 (dm )

+ (−1)

m



C m−1 j,m−1

= (−1) (m − 1)! m

j=1

m

di .

(2.12)

i=1

A set of m − 2 equations (2.11) together with Eq. (2.5) coincide with the Herzog– Kühl equations [18]1 for the Betti numbers of graded Cohen–Macaulay modules of codimension m − 1; a unity in m − 1 stands for the Krull dimension of the semigroup ring associated with S (dm ). However, Eq. (2.12) has not been discussed earlier. Another kind of identities, also never discussed in literature, appears when we keep in mind relationships among the generators di ∈ dm . Namely, if there exists a set q (dm ) such that       q dm ⊂ {d1 , . . . , dm } , q dm := {di | q | di } , ωq = #q dm , (2.13) then there hold another type of identities, which are not polynomials. Theorem 2 Let the numerical semigroup S (dm ) be given with its Hilbert series H (dm ; z) in accordance with (2.2, 2.3). Then for every q and n such that q | di , 1 < q ≤ max {d1 , . . . , dm } and gcd(n, q) = 1, the following quasipolynomial identities hold: β1 (dm )

C rj,1 exp

j=1

 β2    (dm ) 2π n 2π n i C rj,2 exp i C j,1 − C j,2 + · · · q q j=1

1 I am grateful to B. Sturmfels who brought this fact to my attention.

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  2π n C j,m−1 = 0 C rj,m−1 exp i q

βm−1 (dm )

+ (−1)

m

 j=1

where r = 1, . . . , ωq − 1. If r = 0, then the following trigonometric identity holds: β1 (dm ) j=1

 β2    (dm ) 2π n 2π n exp i exp i C j,1 − C j,2 + · · · q q j=1

   2π n C j,m−1 = 1. exp i q

βm−1 (dm )

+ (−1)m

(2.14)

j=1

Theorem 2 leads to another statement regardless of the inner relations between the generators di . Indeed, by consequence of (2.14) and due to the fact that the generating set dm is minimal, we can replace q by any generator di , gcd(n, di ) = 1 in (2.14). In Sect. 5, we discuss special cases of numerical semigroups, where identities (2.11, 2.12) become unexpectedly simple (see Corollary 1 for CI).

3 Restricted partition functions W (s, dm ) In partition theory, there exists a generating function which is rather similar to the rational Rep (2.2) of the Hilbert series H (dm ; z). This is the Molien function M(dm ; z) [1], m ∞      1 M dm ; z = = W s, dm z s , (3.1) d 1−z i i=1

s=0

that generates a restricted partition function W (s, dm ) which gives a number of partitions of s ≥ 0 into positive integers {d1 , . . . , dm }, each not exceeding s, and vanishes, if such partition does not exist. We present several basic properties of W (s, dm ), which will become useful in the study of H (dm ; z) in Sect. 4. More details on W (s, dm ) may be found in Appendix 1. According to Proposition 4.4.1, [29], the function W (s, dm ) is a quasipolynomial of degree m − 1,         W s, dm = K 1 s, dm s m−1 + K 2 s, dm s m−2 + · · · + K m−1 s, dm s   + K m s, dm , (3.2) where K j (s, dm ) is a periodic function with integer period τ j dividing T = lcm(d1 , . . . , dm ). According to Schur’s theorem (see [31], Theorem 3.15.2), K 1 (s, dm ) is independent of s,   K 1 s, dm =

1 , πm = di . (m − 1)! πm m

i=1

123

(3.3)

Restricted Partition Functions and Identities

The other K j (s, dm ), 1 < j ≤ m, can be calculated by a computational algorithm [8]. Although a variable s in (3.1, 3.2) is assumed to be integer, Rep (3.2) can be extended to the real values of s. However, such extension it depends on a choice  is not unique,   of a set of periodic functions, usually sin 2πT k s and cos 2πT k s , e.g., [20] W (s, {3, 5, 7}) s2 s 74 2 2π s = + + + cos 210  14 315 9 3   

 π 2 2π 2 8 2π s 4π s sin + sin + cos cos 25 5 5 5 5    

π 2 6π s 2π 2 2π s 4π s 2 6π cos + 2 sin + 2 sin sin sin − √ sin 7 7 7 7 7 7 7 7    2 2π s 4π 4π s 3π 6π s 2 2π cos + sin cos + 2 sin sin + √ sin . (3.4) 7 7 7 7 7 7 7 7 Another way to deal with quasipolynomial W (s, dm ), which does not involve computational algorithms, is to decompose  it into a finite  number of terms, each containing  a single periodic function sin 2πT k s or cos 2πT k s as in (3.4). Because of this periodicity, Sylvester [30] refers to them as waves depending on all roots of unity. 3.1 Sylvester waves and their quasipolynomial representations Following [30], decompose W (s, dm ) into a finite number of quasipolynomials Wq (s, dm ), each containing a single q-periodic function associated with the qth root ξ(q) of unity, 

W s, d

m



=

m max d

  Wq s, dm , max dm = max{d1 , . . . , dm },

(3.5)

q=1, q|di

where summation runs over all distinct factors of m generators di . Following [8], the quasipolynomials Wq (s, dm ) are called the Sylvester waves. Sylvester proved [30] that Wq (s, dm ) is a residue at a point z = 0 of a function Fq (s, z), Fq (s, z) =

 1≤n 1, reads [26]   Wq s, dm =

1 (ωq − 1)! πωq



  W q,n s, dm , πωq =

di (3.8)

di ∈q (dm )

1≤nk=1

(e j + er )k +

m−1  j>k>l=1

⎛

m−1 

(e j + er + el )k − · · · + (−1)m ⎝

⎞k e j ⎠ = 0.

j=1

(5.2)

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Restricted Partition Functions and Identities

Identity (5.2) looks very ’combinatorial’ and, indeed, for k = 1 it can be verified trivially, m−1  j=1

ej −

m−1 

(e j + er ) + · · ·

j>k=1

    m−2 + (−1)m E = 1 − + · · · + (−1)m E = (1 − 1)m−2 E = 0. 1

However, for 1 < k < m − 1 such a straightforward way of verification is not easy. In Appendix 2 we give its proof based on the inclusion–exclusion principle applied to the set of tuples of length k comprising elements of m − 1 different colors.2 Calculate the l.h.s. in (5.2) when k = m − 1 and obtain the following identity, m−1  j=1

em−1 − j

m−1 

(e j + er )m−1 + · · · + (−1)m E m−1

j>k=1

= (−1)m (m − 1)!

m−1

ei .

(5.3)

i=1

Combining the last identity with the second identity (2.12) in Theorem 1, we obtain Corollary 1 Let the CI semigroup be given with its Hilbert series H (dm ; z) in accordance with (2.2, 2.10). Then the following identity holds, m−1

ei = πm .

(5.4)

i=1

The entire set of the m − 1 polynomial identities of Theorem 1 is reduced here to identity (5.4). The quasipolynomial relations of Theorem 2 together with identity (5.4) present m − 1 nonlinear Diophantine equations for m − 1 positive integer variables ei . As an example, we present two CI semigroups generated by four elements [7]: (a) (8, 9, 10, 12) and (b) (10, 14, 15, 21), where Corollary 1 is satisfied, (1 − z 18 )(1 − z 20 )(1 − z 24 )    , Ha (z) =  1 − z 8 1 − z 9 1 − z 10 1 − z 12 (1 − z 30 )(1 − z 35 )(1 − z 42 )     . Hb (z) =  1 − z 10 1 − z 14 1 − z 15 1 − z 21 Corollary 2 Let the CI semigroup be given with its Hilbert series H (dm ; z) in accordance with (2.2, 2.10). Then the Frobenius number F (dm ) has a lower bound   √ F dm ≥ (m − 1) m−1 πm − σ1 .

(5.5)

2 Combinatorial proof of (5.2) was kindly communicated to me by R. Pinchasi.

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Proof Consider the CI semigroup and write its Frobenius number in accordance with m−1 ei − σ1 . Making use of algebraic inequality [15] Z 1 /(m − (2.7, 2.10): F (dm ) = i=1 m−1 √ m−1 m−1 Z m−1 for symmetric polynomials Z 1 = i=1 ei and Z m−1 = !i=1 ei and 1) ≥ m−1 m−1 m m substituting it into the last formula for F (d ) we obtain, F (d ) ≥ (m−1) i=1 ei −  σ1 . Combining this inequality with identity (5.4) from Corollary 1 we arrive at (5.5).  Formula (5.5) gives a pretty good lower bound for small di , e.g., F(8, 9, 10, 12) = 23 > 22.559, F(10, 14, 15, 21) = 47 > 45.991. It is interesting to compare (5.5) with the lower bound [19] for the Frobenius number of a nonsymmetric numerical semigroup of general kind,   F dm ≥

"

m−1

√ (m − 1)! m−1 πm − σ1 .

(5.6)

5.2 Symmetric numerical semigroups This kind of semigroups is of high importance in commutative algebra due to Kunz’s theorem [22], which asserts that a graded polynomial subring associated with S (dm ) is Gorenstein if a semigroup is symmetric. For short, denote Q (dm ) = deg Q (dm ; z) and define the following combination of the Betti numbers, #m $         , (5.7) B dm = −1 + β1 dm − β2 dm + · · · + (−1)μ βμ−1 dm , μ = 2 where a is the floor function which gives the largest integer less than or equal to a. Then, by Theorem 1 for symmetric semigroups with even edim = 2m, the following identity holds for 1 ≤ k ≤ 2m − 1, 



βr d2m 

 m−1  k  

  k 2m r k 2m − − C j,r C j,r − Q d Q d (−1) r =1

j=1

= (2m − 1)! π2m δk,2m−1 .

(5.8)

In the case of k = 1, the last identity reads   β1 d2m



  β2 d2m

C j,1 −

j=1



C j,2 + · · ·

j=1  2m  βm−1 d

+ (−1)m



C j,m−1 =

j=1

1 2m  2m  . B d Q d 2

(5.9)

By Theorem 1 for symmetric semigroups with odd edim = 2m + 1, it holds for 1 ≤ k ≤ 2m,



Qk d2m+1 +

m−1  r =1

123

  βr d2m+1 

(−1)r

 j=1

 k  

C kj,r + Q d2m+1 − C j,r

Restricted Partition Functions and Identities   βm d2m+1

+ (−1)m



C kj,m + (2m)! π2m+1 δk,2m = 0.

(5.10)

j=1

In the case of k = 1, the last identity reads   βm d2m+1

(−1)m





 C j,m = B d2m+1 Q d2m+1 .

(5.11)

j=1

    Identity (5.11) leads to two inequalities, B d4m+1 > 0 and B d4m+3 < 0. Substituting (5.7) into both of them and combining the results together, we get  2m+1 m−1 rβ (−1) d > 0. The last inequality is rather trivial and may be derived m−1−r r =0 if we combine (2.5) and (2.9).

  5.2.1 Symmetric semigroups S d4 (not CI)   Bresinsky [3] has shown that symmetric semigroups S d4 , which are not CI, have   β1 d4 = 5. Denote basic polynomials Ik = 5j=1 C kj,1 and obtain by (5.8, 5.9) two polynomial identities,

 1 8I3 − 6I2 I1 + I13 = 24π4 , Q d4 = I1 . 2

(5.12)

We illustrate identities (5.12) by presenting the numerator of the Hilbert series of the symmetric semigroup S ({5, 6, 7, 8}) which is not CI [7], Q(z) = 1 − z 12 − z 13 − z 14 − z 15 − z 16 + z 19 + z 20 + z 21 + z 22 + z 23 − z 35 . The quasipolynomial identities of Theorem 2 are also satisfied. Recently, identity (5.12) was used [13] to derive a lower bound for the Frobenius number.   Corollary 3 [13] Let the symmetric (not CI) semigroup S d4 be given. Then the Frobenius number has a lower bound

 " (5.13) F d4 ≥ 3 25π4 − σ1 .

  5.2.2 Symmetric semigroups S d5 (not CI) The problem of admissible values of β1 (dm ), m ≥ 5, for symmetric semigroups (not CI) or their upper bounds, is still open. In thissection, we study the polynomial identities   of Theorem 1 for m = 5 and arbitrary β1 d5 ≥ 5. The last inequality holds since S d5 is not CI. Skip for short the notation d5 in the Betti numbers and denote the invariants Ir,k = βr k j=1 C j,r , β2 = 2(β1 − 1), r = 1, 2, and 1 ≤ k ≤ 4. By (5.10) we obtain four different polynomial identities,

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L. G. Fel

 I2,1 = χ Q d5 , χ = β1 − 1,   I2,1 (2I1,1 − I2,1 ) + χ I2,2 − 2I1,2 = 0, 2 I2,1 (3I1,1 − I2,1 ) − 3χ I2,1 I1,2 + χ 2 I2,3 = 0, 3 2 (4I1,1 − I2,1 ) − 6χ I2,1 I1,2 I2,1   2 3 + 4χ I2,1 I1,3 + χ I2,4 − 2I1,4 = 24χ 3 π5 .

(5.14)

  Bresinsky [4] has proven that if a symmetric semigroup S d5 , which is not CI, has any four  generators satisfying di + d j = dk + dl , where i, j, k, l, ∈ {1, 2, 3, 4, 5}, then β1 d4 ≤ 13. He has found numerically such semigroup 19, 23, 29, 31, 37 where the upper bound is attainable. Its Betti numbers read as follows: β1 = 13, β2 = 24, β3 = 13, β4 = 1. The numerator Q(z) of its Hilbert series was calculated by the author, Q(z) = 1 − z 60 − z 69 − z 75 − z 77 − z 81 − z 85 − z 87 − z 93 − z 95 + z 98 − z 99 + z 100 − z 103 + z 104 − z 105 + 2z 106 + z 108 + z 110 − z 111 + z 112 + z 114 + z 116 + 2z 118 + 2z 122 + z 124 + z 126 + z 128 − z 129 + z 130 + z 132 + 2z 134 − z 135 + z 136 − z 137 + z 140 − z 141 + z 142 − z 145 − z 147 − z 153 − z 155 − z 159 − z 163 − z 165 − z 171 − z 180 + z 240 . Straightforward calculation shows that the polynomial identities (5.14) are satisfied as well as the quasipolynomial identities of Theorem 2.

6 Concluding Remarks The preprint of this article was posted [12] on arXiv.org in the end of 2009 and presented in the Numerical semigroups conference (Granada, 2010) but has never been published. Later its main results were posed in the open problem session in a conference (Catania, 2011) in order to find an algebraic proof of Theorems 1 and 2. It was done in [21] following the proof of the Herzog–Kühl theorem [18]. The identities in Theorems 1 and 2 have been left unchanged. In addition to the combinatorial proof of Theorems 1 and 2, the preprint contained an application of identities to the different kinds of semigroups (symmetric, complete intersection (CI), symmetric (not CI) with edim = 4, 5, nonsymmetric with edim = 3 , etc). These results are of interest by themselves; however, they were not delivered to a wide audience. That is why I decided to publish this work in the present version which slightly differs from the initial preprint [12] by new statements in Sect. 4 (Theorem 3) and in Sect. 5 (Corollaries 2, 3). Acknowledgements I am grateful to R. Fröberg for encouraging me to publish this article. I thank A. Juhasz, R. Pinchasi and B. Sturmfels for useful discussions and an unknown referee for helpful comments. The research was partly supported by the Kamea Fellowship.

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Appendix 1: Sylvester waves In this Appendix, we present detailed formulas for the Sylvester waves which specify the coefficients in the polynomial Reps (3.13–3.15). We start with a brief explanation of “symbolic formulas” used in Sect. 3.1. Its origin dates back to Sylvester pointed out a similarity between binomial expansion n  n−kwhich n k and the relation for Bernoulli polynomials B (x) = x y (y + x)n = n k=0 k n  n k k=0 k bn−k x , where bn denote the Bernoulli numbers. By visual similarity, write the last relation in symbolic form Bn (x) = (b + x)n , where after binomial expansion the symbolic powers (bn−k x k ) are converted into x k multiplied by Bernoulli numbers bn−k , i.e., (bn−k x k ) → bn−k x k . Symbolic calculus becomes popular in combinatorics and partition theory after the paper [25].

The first Sylvester wave Performing a binomial expansion for W1 (s, dm ) in (3.7) and comparing it with (3.16), we obtain 1 Lm (s)

=

m−1  r =0

 m−1 fr · s m−1−r , r

 fr = σ1 +

m 

r B di

.

(7.1)

i=1

m Denote σk = i=1 dik and make use of Bernoulli numbers, B0 = 1, B1 = − 21 , B2 = 1 6 , . . ., B2k+1 = 0, k ≥ 1, to perform a straightforward calculation in (7.1),

 1 2 σ2  σ1 2 − = − σ , f σ σ , 3 2 22 1  3 23 1 σ2 1 2σ4 , f 4 = 4 σ14 − 2σ12 σ2 + 2 + 2 3 15   5σ22 10σ12 σ2 2σ4 σ1 4 + + , etc. f 5 = 5 σ1 − 3 3 3 2

f 0 = 1,

σ1 f1 = , 2 

f2 =

(7.2)

By comparing (3.16), (3.13), and (7.1), we conclude   m−1 f k−1 , 1 = 1 .

k = k−1

(7.3)

The higher Sylvester waves, q ≥ 2 The symbolic formula for the partial wave Wq,n (s, dm ) reads (see [26], formula (28)), −sn ξ(q)

 di n i=ωq +1 1 − ξ(q)

  Wq,n s, dm =  m

123

L. G. Fel

⎛ × ⎝s + σ1 +

ωq 

B di +

i=1

m 

⎞ωq −1 

di n di ⎠ H ξ(q) .

(7.4)

i=ωq +1

 r



di n di in (7.4) has to be converted Symbolic binomial expansion of the powers H ξ(q)





 r di n di n into the generator’s powers multiplied by the numbers Hr ξ(q) , i.e., H ξ(q) di →

 di n ; they generalize the values of the Euler polynomials Er (x) [26], Er (0) = dir Hr ξ(q) Hr (−1). These numbers were introduced by Frobenius [14] and Carlitz [5] as the values di n , where Hn (x) itself comes by power expansion of the rational function Hn (x) at x = ξ(q) of its generating function,

  tn 1−x =1+ Hn (x) , where Hn x −1 = (−1)n x Hn (x) , t e −x n! ∞

H2n (−1) = 0,

n=1

(7.5) and for x = 1 the four first rational functions Hn (x) read 1 x +1 , H2 (x) = , x −1 (x − 1)2 x 3 + 11x 2 + 11x + 1 H4 (x) = . (x − 1)4

H1 (x) =

H3 (x) =

x 2 + 4x + 1 , (x − 1)3

Consider the representation (3.11) and rewrite it as follows, −sn sn + W m Lm (s) = Wq,n (s, dm ) ξ(q) q,−n (s, d ) ξ(q) , q,n −sn sn m m . i Mm (s) = Wq,n (s, d ) ξ(q) − Wq,−n (s, d ) ξ(q) q,n

q,n

(7.6)

q,n

On the other hand, Lm (s) and Mm (s) are defined by their polynomial expansions (3.14) q,n q,n and (3.15), respectively. We can calculate coefficients Lr and Mr in these expansions by comparing (3.14) and (3.15) with equalities (7.6) after substituting in the latter a q,n q,n binomial expansion of symbolic Rep (7.4). The final formulas for Lr and Mr are given in the author’s preprint [12], section 4.3, formulas (4.19, 4.20) and look cumbersome. However, according to Sect. 4.2.3, to derive the quasipolynomial identities for degrees of q,n q,n syzygies in the Hilbert series (2.2), we have to know only their first pair, L1 and M1 . That is why in this Appendix we give their short derivation. Perform a necessary substitution of the binomial expansion of symbolic Rep (7.4) into equalities (7.6) and preserve the terms of the highest order in s, i.e., s ωq −1 . Comparing them with polynomial expansions (3.14) and (3.15), we obtain,

(7.7)

1 1

−

. di n −di n m i=ωq +1 1 − ξ(q) i=ωq +1 1 − ξ(q)

(7.8)

=  m

q,n

=  m

i M1

123

1 1

+

, di n −di n m 1 − ξ 1 − ξ i=ωq +1 i=ωq +1 (q) (q)

q,n

L1

Restricted Partition Functions and Identities q,n

Find requirements for the parameters σ1 , n, q, ωq , and m, providing L1 = 0 and q,n M1 = 0 separately. Simple analysis of formulas (7.7, 7.8) gives a list of selection rules for these coefficients and explains why some of them disappear in (3.4). Lemma 1 For k ∈ N the following identities hold q,n

L1

= 0,

q,n

M1

= 0,

σ1 · n or = k, m − ωq = 1 (mod 2), q 1 σ1 · n = k + , m − ωq = 0 (mod 2), q 2 σ1 · n if = k, m − ωq = 0 (mod 2), or q 1 σ1 · n = k + , m − ωq = 1 (mod 2). q 2

if

Proof Keeping in mind that the product in (7.7, 7.8) is taken over generators di , which are m ξ di n . Making use of not divisible by q, multiply the second term in (7.7, 7.8) by i=ω q +1 (q) ωq di n ξ(q) = 1 for generators di , which are divisible by q, rewrite formulas (7.7, identity i=1 7.8) as follows, σ1 n σ1 n 1 + (−1)m−ωq ξ(q) 1 − (−1)m−ωq ξ(q)

 , i Mq,n

, =  = m 1 di n di n m i=ωq +1 1 − ξ(q) i=ωq +1 1 − ξ(q)   σ1 n σ1 n ξ(q) = exp 2iπ , q

q,n

L1

(7.9)

where σ1 encompasses the both kind of generators di . Setting the numerators in (7.9) to zero, we finish the proof of Lemma 1.   Proposition 1 in Sect. 3.1 is an important consequence of Lemma 1 for the study of the q,n q,n relationships between degrees C j,i of syzygies in Sect. 4.2. It states that L1 and M1 cannot vanish simultaneously.

Appendix 2: Combinatorial proof of identities (5.2) and (5.3) Consider n sets of elements a j,r ∈ Er , #Er = er , 1 ≤ r ≤ n, of different colors and construct a tuple ak of length k which is composed of elements a j,r . Denote by Pi a set of all ak -tuples which do not contain any element of the ith color, i.e., Pi = {ak | a j,r ∈ ak , r = i}, #Pi = (E − ei )k ,

E=

n 

ej .

(7.10)

j=1

Consider an intersection of two such sets Pi and Pl which do not contain any element of the ith and lth colors, Pi ∩ Pl = {ak | a j,r ∈ ak , r = i, r = l}, # {Pi ∩ Pl } = (E − ei − el )k . (7.11)

123

L. G. Fel

Continuing to build intersections of three and more sets Pi write by the inclusion–exclusion principle an identity for their cardinalities, #

n %

Pi =

i=1

n 

(E − ei )k −

i=1

− (−1)n

n 

(E − ei − el )k + · · ·

i>l=1 n 

(ei + el )k + (−1)n

i>l=1

n 

eik ,

(7.12)

i=1

&n &n where i=1 Pi = {ak | a j,r ∈ ak , 1 ≤ r ≤ n}. If k < n then # i=1 Pi = E k and therefore n 

eik −

i=1

n 

(ei + el )k + · · · + (−1)n

i>l=1

i=1

ein −

n 

(E − ei )k − (−1)n E k = 0 .

(7.13)

i=1

In the case k = n, we have # we obtain n 

n 

&n

i=1 Pi

= E n − n!

n

(ei + el )n + · · · + (−1)n

i>l=1

= −(−1)n n!

i=1 ei .

n 

Inserting the latter into (7.12),

(E − ei )n − (−1)n E n

i=1 n

ei .

(7.14)

i=1

Substituting n = m − 1 into (7.13, 7.14) we arrive at (5.2, 5.3), respectively.

References 1. Andrews, G.E.: The theory of partitions. Encyclopedia of Mathematics and Its Applications, vol. 2. Addison, Reading, MA (1976) 2. Aicardi, F., Fel, L.G.: Gaps in nonsymmetric numerical semigroups. Israel J. Math. 175, 85–112 (2010) 3. Bresinsky, H.: Symmetric semigroups of integers generated by 4 elements. Manuscripta Math. 17, 205–219 (1975) 4. Bresinsky, H.: Monomial Gorenstein ideals. Manuscripta Math. 29, 159–181 (1979) 5. Carlitz, L.: Eulerian numbers and polynomials of higher order. Duke Math. J. 27, 401–423 (1960) 6. Delgado, M., García-Sánchez, P.A., J. Morais: A GAP Package on Numerical Semigroups 7. Delorme, C.: Sous-Monoïdes d’Intersection Compléte de N, Ann. Scient. de l’École Normale Supérieure, Sér. 4, 9(1), 145–154 (1976) 8. Fel, L.G., Rubinstein, B.Y.: Sylvester waves in the Coxeter groups. Ramanujan J. 6(3), 307–329 (2002) 9. Fel, L.G.: Frobenius problem for semigroups S (d1 , d2 , d3 ), Funct. Anal. Math. 1(2), 119–157 (2006); http://xxx.lanl.gov/abs/math/0409331 (2004) 10. Fel, L.G.: Symmetric numerical semigroups generated by Fibonacci and Lucas triples. INTEGERS 9, 106–116 (2009) 11. Fel, L.G.: Duality relation for the Hilbert series of almost symmetric numerical semigroups. Israel J. Math. 185, 413–444 (2011) 12. Fel, L.G.: New Identities for Degrees of Syzygies in Numerical Semigroups. arXiv:0912.5192 13. Fel, L.G.: On Frobenius numbers for symmetric (not complete intersection) semigroups generated by four elements. Semigroup Forum 93, 423–426 (2016)

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