IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Some comments on denumerant of numerical3–semigroups

IMNS 2014, Cortona

Francesc Aguiló-Gost,Dept. MA-IV, Univ. Politècnica de Catalunya, Barcelona

Pedro A. García-Sánchez,Depto. de Álgebra, Univ. de Granada, Granada

David Llena,Depto. de Matemáticas, Univ. de Almería, Almería

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Introduction

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Definitions and notation.

Given a1, . . . , an ∈ N, with 1 ≤ a1 < a2 < ... < an andgcd(a1, ..., an) = 1, the numerical n–semigrup S = 〈a1, . . . , an〉 isdefined by

〈a1, . . . , an〉 = {x1a1 + · · ·+ xnan : (x1, ..., xn) ∈ Nn}.

Given m ∈ S, a vector (x1, ..., xn) ∈ Nn is a factorization of m ifx1a1 + · · ·+ xnan = m.

F(m, 〈a1, ..., an〉) = {(x1, ..., xn) ∈ Nn : x1a1 + · · ·+ xnan = m},

d(m, 〈a1, ..., an〉) = |F(m, 〈a1, ..., an〉)| .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Definitions and notation.

Given a1, . . . , an ∈ N, with 1 ≤ a1 < a2 < ... < an andgcd(a1, ..., an) = 1, the numerical n–semigrup S = 〈a1, . . . , an〉 isdefined by

〈a1, . . . , an〉 = {x1a1 + · · ·+ xnan : (x1, ..., xn) ∈ Nn}.

Given m ∈ S, a vector (x1, ..., xn) ∈ Nn is a factorization of m ifx1a1 + · · ·+ xnan = m.

F(m, 〈a1, ..., an〉) = {(x1, ..., xn) ∈ Nn : x1a1 + · · ·+ xnan = m},

d(m, 〈a1, ..., an〉) = |F(m, 〈a1, ..., an〉)| .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Definitions and notation.

Given a1, . . . , an ∈ N, with 1 ≤ a1 < a2 < ... < an andgcd(a1, ..., an) = 1, the numerical n–semigrup S = 〈a1, . . . , an〉 isdefined by

〈a1, . . . , an〉 = {x1a1 + · · ·+ xnan : (x1, ..., xn) ∈ Nn}.

Given m ∈ S, a vector (x1, ..., xn) ∈ Nn is a factorization of m ifx1a1 + · · ·+ xnan = m.

F(m, 〈a1, ..., an〉) = {(x1, ..., xn) ∈ Nn : x1a1 + · · ·+ xnan = m},

d(m, 〈a1, ..., an〉) = |F(m, 〈a1, ..., an〉)| .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Definitions and notation.

Given a1, . . . , an ∈ N, with 1 ≤ a1 < a2 < ... < an andgcd(a1, ..., an) = 1, the numerical n–semigrup S = 〈a1, . . . , an〉 isdefined by

〈a1, . . . , an〉 = {x1a1 + · · ·+ xnan : (x1, ..., xn) ∈ Nn}.

Given m ∈ S, a vector (x1, ..., xn) ∈ Nn is a factorization of m ifx1a1 + · · ·+ xnan = m.

F(m, 〈a1, ..., an〉) = {(x1, ..., xn) ∈ Nn : x1a1 + · · ·+ xnan = m},

d(m, 〈a1, ..., an〉) = |F(m, 〈a1, ..., an〉)| .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Definitions and notation

Given N ∈ S = 〈a1, ..., an〉, the Apéry set Ap(S,N) is

Ap(S,N) = {s ∈ S : s−N 6∈ S}.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Know results

Generating function of d(m, 〈a1, ..., an〉) (Sylvester, 1882):

φ(z) =1

(1− za1)(1− za2) · · · (1− zan).

Assymptotic behaviour (Schur, 1926):

lim supm→∞

d(m, 〈a1, ..., an〉)mn−1

a1a2···an(n−1)!= 1.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Know results

Generating function of d(m, 〈a1, ..., an〉) (Sylvester, 1882):

φ(z) =1

(1− za1)(1− za2) · · · (1− zan).

Assymptotic behaviour (Schur, 1926):

lim supm→∞

d(m, 〈a1, ..., an〉)mn−1

a1a2···an(n−1)!= 1.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Know results

Denumerant formula for n = 2 (Popoviciu, 1953)

d(m, 〈p, q〉) = m+ pf(m) + qg(m)

lcm(p, q)− 1

where f(m) ≡ −mp−1(mod q) and g(m) ≡ −mq−1(mod p).

Recursive denumerant formulae for n = 3, 4 (Ehrhart, 1967).

For n = 3, an O(ab) algorithm for computing d(m, 〈a, b, c〉) wasgiven by Lisoněk 1995.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Know results

Denumerant formula for n = 2 (Popoviciu, 1953)

d(m, 〈p, q〉) = m+ pf(m) + qg(m)

lcm(p, q)− 1

where f(m) ≡ −mp−1(mod q) and g(m) ≡ −mq−1(mod p).

Recursive denumerant formulae for n = 3, 4 (Ehrhart, 1967).

For n = 3, an O(ab) algorithm for computing d(m, 〈a, b, c〉) wasgiven by Lisoněk 1995.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definitions and notationKnown results

Know results

Denumerant formula for n = 2 (Popoviciu, 1953)

d(m, 〈p, q〉) = m+ pf(m) + qg(m)

lcm(p, q)− 1

where f(m) ≡ −mp−1(mod q) and g(m) ≡ −mq−1(mod p).

Recursive denumerant formulae for n = 3, 4 (Ehrhart, 1967).

For n = 3, an O(ab) algorithm for computing d(m, 〈a, b, c〉) wasgiven by Lisoněk 1995.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Distribution of 3–factorizations

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Every numerical 3–semigroup S = 〈a, b, c〉 has related a planeL-shape H that encodes the set Ap(S, c).

An L-shape is denotedby the lengths of her sides H = L(l, h, w, y).

Example: Ap(〈5, 7, 11〉, 11) = {0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 24}

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H = L(5, 3, 2, 2)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Every numerical 3–semigroup S = 〈a, b, c〉 has related a planeL-shape H that encodes the set Ap(S, c). An L-shape is denotedby the lengths of her sides H = L(l, h, w, y).

Example: Ap(〈5, 7, 11〉, 11) = {0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 24}

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H = L(5, 3, 2, 2)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Every numerical 3–semigroup S = 〈a, b, c〉 has related a planeL-shape H that encodes the set Ap(S, c). An L-shape is denotedby the lengths of her sides H = L(l, h, w, y).

Example: Ap(〈5, 7, 11〉, 11) = {0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 24}

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H = L(5, 3, 2, 2)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Every numerical 3–semigroup S = 〈a, b, c〉 has related a planeL-shape H that encodes the set Ap(S, c). An L-shape is denotedby the lengths of her sides H = L(l, h, w, y).

Example: Ap(〈5, 7, 11〉, 11) = {0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 24}

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H = L(5, 3, 2, 2)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Every numerical 3–semigroup S = 〈a, b, c〉 has related a planeL-shape H that encodes the set Ap(S, c). An L-shape is denotedby the lengths of her sides H = L(l, h, w, y).

Example: Ap(〈5, 7, 11〉, 11) = {0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 24}

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H = L(5, 3, 2, 2)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Also 3–factorizations (x, y, z) ∈ F(m, 〈a, b, c〉) follow thisL-shaped plane distribution.

Given a related L-shape H, each m ∈ 〈a, b, c〉 has a unique basicfactorization (x0, y0, z0) with (x0, y0) ∈ H.

Example: F(87, 〈5, 7, 11〉)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Also 3–factorizations (x, y, z) ∈ F(m, 〈a, b, c〉) follow thisL-shaped plane distribution.

Given a related L-shape H, each m ∈ 〈a, b, c〉 has a unique basicfactorization (x0, y0, z0) with (x0, y0) ∈ H.

Example: F(87, 〈5, 7, 11〉)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

Also 3–factorizations (x, y, z) ∈ F(m, 〈a, b, c〉) follow thisL-shaped plane distribution.

Given a related L-shape H, each m ∈ 〈a, b, c〉 has a unique basicfactorization (x0, y0, z0) with (x0, y0) ∈ H.

Example: F(87, 〈5, 7, 11〉)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

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{(2, 0, 7)}

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

0

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{(2, 0, 7), (0, 3, 6)}

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

0

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{(2, 0, 7), (0, 3, 6), (5, 1, 5), (3, 4, 4), (1, 7, 3)}

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

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40

47

54

61

68

75

82

89

96

103

110

117

124

131

138

45

52

59

66

73

80

87

94

101

108

115

122

129

136

143

50

57

64

71

78

85

92

99

106

113

120

127

134

141

148

55

62

69

76

83

90

97

104

111

118

125

132

139

146

153

60

67

74

81

88

95

102

109

116

123

130

137

144

151

158

65

72

79

86

93

100

107

114

121

128

135

142

149

156

163

70

77

84

91

98

105

112

119

126

133

140

147

154

161

168

75

82

89

96

103

110

117

124

131

138

145

152

159

166

173

80

87

94

101

108

115

122

129

136

143

150

157

164

171

178

85

92

99

106

113

120

127

134

141

148

155

162

169

176

183

90

97

104

111

118

125

132

139

146

153

160

167

174

181

188

95

102

109

116

123

130

137

144

151

158

165

172

179

186

193

{(2, 0, 7), (0, 3, 6), (5, 1, 5), (3, 4, 4), (1, 7, 3), (8, 2, 3), (13, 0, 2),

(6, 5, 2), (4, 8, 1), (2, 11, 0)}

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Plane distribution

0

7

14

21

28

35

42

49

56

63

70

77

84

91

98

5

12

19

26

33

40

47

54

61

68

75

82

89

96

103

10

17

24

31

38

45

52

59

66

73

80

87

94

101

108

15

22

29

36

43

50

57

64

71

78

85

92

99

106

113

20

27

34

41

48

55

62

69

76

83

90

97

104

111

118

25

32

39

46

53

60

67

74

81

88

95

102

109

116

123

30

37

44

51

58

65

72

79

86

93

100

107

114

121

128

35

42

49

56

63

70

77

84

91

98

105

112

119

126

133

40

47

54

61

68

75

82

89

96

103

110

117

124

131

138

45

52

59

66

73

80

87

94

101

108

115

122

129

136

143

50

57

64

71

78

85

92

99

106

113

120

127

134

141

148

55

62

69

76

83

90

97

104

111

118

125

132

139

146

153

60

67

74

81

88

95

102

109

116

123

130

137

144

151

158

65

72

79

86

93

100

107

114

121

128

135

142

149

156

163

70

77

84

91

98

105

112

119

126

133

140

147

154

161

168

75

82

89

96

103

110

117

124

131

138

145

152

159

166

173

80

87

94

101

108

115

122

129

136

143

150

157

164

171

178

85

92

99

106

113

120

127

134

141

148

155

162

169

176

183

90

97

104

111

118

125

132

139

146

153

160

167

174

181

188

95

102

109

116

123

130

137

144

151

158

165

172

179

186

193

{(2, 0, 7), (0, 3, 6), (5, 1, 5), (3, 4, 4), (1, 7, 3), (8, 2, 3), (13, 0, 2),

(6, 5, 2), (4, 8, 1), (2, 11, 0), (11, 3, 1), (16, 1, 0), (9, 6, 0)}

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Denumerant sums

Theorem (A., García-Sánchez 2010)Given m ∈ 〈a, b, c〉 and H = L(l, h, w, y), set δ = la−yb

c ,θ = −wa+hb

c and the basic factorization (x0, y0, z0) of m in H.

Define Sk and Tk as

Sk =

⌊y0+k(h−y)

y

⌋δ = 0,⌊

z0−k(δ+θ)δ

⌋y = 0,

min{⌊y0+k(h−y)

y

⌋,⌊z0−k(δ+θ)

δ

⌋} δy 6= 0,

Tk =

⌊x0+k(l−w)

w

⌋θ = 0,⌊

z0−k(δ+θ)θ

⌋w = 0,

min{⌊x0+k(l−w)

w

⌋,⌊z0−k(δ+θ)

θ

⌋} θw 6= 0,

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Denumerant sums

Theorem (A., García-Sánchez 2010)Given m ∈ 〈a, b, c〉 and H = L(l, h, w, y), set δ = la−yb

c ,θ = −wa+hb

c and the basic factorization (x0, y0, z0) of m in H.Define Sk and Tk as

Sk =

⌊y0+k(h−y)

y

⌋δ = 0,⌊

z0−k(δ+θ)δ

⌋y = 0,

min{⌊y0+k(h−y)

y

⌋,⌊z0−k(δ+θ)

δ

⌋} δy 6= 0,

Tk =

⌊x0+k(l−w)

w

⌋θ = 0,⌊

z0−k(δ+θ)θ

⌋w = 0,

min{⌊x0+k(l−w)

w

⌋,⌊z0−k(δ+θ)

θ

⌋} θw 6= 0,

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Plane distributionDenumerant sums

Denumerant sums

then

d(m, 〈a, b, c〉) = 1 +

⌊z0δ + θ

⌋+

b z0δ+θc∑k=0

(Sk + Tk)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ and S− sums

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ and S− sums

We define the following discrete sums

S+(s, t, q,N) =N∑k=0

⌊s+ kt

q

⌋and S−(s, t, q,N) =

N∑k=0

⌊s− ktq

⌋when 0 ≤ s, t < q.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ sums

Consider f(x) = b s+xtq c and set Ik by x ∈ Ik ⇔ f(x) = k.

Ik = [xk, xk+1) with xk = kq−st .

5 10 15 20 25 30

2

4

6

8

s = 0, t = 3, q = 10, N = 30

Then, b qt c ≤ |Ik ∩ N| ≤ d qt e (except, possibly, the first and lastintervals).Ik is an hS–type interval if |Ik ∩ N| = d qt e.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ sums

Consider f(x) = b s+xtq c and set Ik by x ∈ Ik ⇔ f(x) = k.Ik = [xk, xk+1) with xk = kq−s

t .

5 10 15 20 25 30

2

4

6

8

s = 0, t = 3, q = 10, N = 30

Then, b qt c ≤ |Ik ∩ N| ≤ d qt e (except, possibly, the first and lastintervals).Ik is an hS–type interval if |Ik ∩ N| = d qt e.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ sums

Consider f(x) = b s+xtq c and set Ik by x ∈ Ik ⇔ f(x) = k.Ik = [xk, xk+1) with xk = kq−s

t .

5 10 15 20 25 30

2

4

6

8

s = 0, t = 3, q = 10, N = 30

Then, b qt c ≤ |Ik ∩ N| ≤ d qt e (except, possibly, the first and lastintervals).Ik is an hS–type interval if |Ik ∩ N| = d qt e.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ sums

Consider f(x) = b s+xtq c and set Ik by x ∈ Ik ⇔ f(x) = k.Ik = [xk, xk+1) with xk = kq−s

t .

5 10 15 20 25 30

2

4

6

8

s = 0, t = 3, q = 10, N = 30

Then, b qt c ≤ |Ik ∩ N| ≤ d qt e (except, possibly, the first and lastintervals).Ik is an hS–type interval if |Ik ∩ N| = d qt e.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

S+ sums

There are three different ways of computing S+, depending onthe cases(1) t | q,(2) t - q and gcd(t, q) = 1,(3) t - q and gcd(t, q) = g > 1.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 1 (t | q)

Set M =⌊s+Ntq

⌋and xM = Mq−s

t , then

S+ =q

t

M(M − 1)

2+M(N − dxMe+ 1).

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 1 (t | q)

Set M =⌊s+Ntq

⌋and xM = Mq−s

t

, then

S+ =q

t

M(M − 1)

2+M(N − dxMe+ 1).

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 1 (t | q)

Set M =⌊s+Ntq

⌋and xM = Mq−s

t , then

S+ =q

t

M(M − 1)

2+M(N − dxMe+ 1).

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Assume t - q.

Set q = qt+ q and s = st+ s with 0 ≤ q, s < t.

Set S = q (M−1)M2 +M(N − dxMe+ 1).

A J ⊂ A set of hS indices is a set of indices in A correspondingto hS–type intervals. Set SJ =

∑k∈J k.

Lemma Assume t - q. Then, Ik is an hS–type interval iff

(s− kq) (mod t) < q.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Assume t - q.

Set q = qt+ q and s = st+ s with 0 ≤ q, s < t.

Set S = q (M−1)M2 +M(N − dxMe+ 1).

A J ⊂ A set of hS indices is a set of indices in A correspondingto hS–type intervals. Set SJ =

∑k∈J k.

Lemma Assume t - q. Then, Ik is an hS–type interval iff

(s− kq) (mod t) < q.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Assume t - q.

Set q = qt+ q and s = st+ s with 0 ≤ q, s < t.

Set S = q (M−1)M2 +M(N − dxMe+ 1).

A J ⊂ A set of hS indices is a set of indices in A correspondingto hS–type intervals. Set SJ =

∑k∈J k.

Lemma Assume t - q. Then, Ik is an hS–type interval iff

(s− kq) (mod t) < q.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Assume t - q.

Set q = qt+ q and s = st+ s with 0 ≤ q, s < t.

Set S = q (M−1)M2 +M(N − dxMe+ 1).

A J ⊂ A set of hS indices is a set of indices in A correspondingto hS–type intervals. Set SJ =

∑k∈J k.

Lemma Assume t - q. Then, Ik is an hS–type interval iff

(s− kq) (mod t) < q.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 2 (t - q and gcd(t, q) = 1)

Consider m0 ≡ q−1s(mod t) and u =⌊M−m0−1

t

⌋. Then,

(1) If m0 = 0 and u ≤ 0, or m0 =M , let K ⊂ {1, . . . ,M − 1} be theset of hS indices. Then S+ = S + SK .

(2) If m0 = 0 and u > 0, take the sets of hS indices J ⊂ {0, . . . , t− 1}and K ⊂ {ut, . . . ,M − 1}. Then S+ = S + uSJ + nt (u−1)u

2 + SK .

(3) If 0 < m0 < M and m0 + t ≥M , take the sets of hS indicesI ⊂ {1, . . . ,m0 − 1} and K ⊂ {m0, . . . ,M − 1}. ThenS+ = S + SI + SK .

(4) If 0 < m0 < M and m0 + t < M , take the sets of hS indicesI ⊂ {1, . . . ,m0 − 1} and J ⊂ {m0,m0 + t− 1}. TakeK ⊂ {m0 + ut, . . . ,M − 1} whenever u > 0 and K = ∅ otherwise(thus SK = 0). Then S+ = S + SI + uSJ + nt (u−1)u

2 + SK .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 2 (t - q and gcd(t, q) = 1)Consider m0 ≡ q−1s(mod t) and u =

⌊M−m0−1

t

⌋.

Then,

(1) If m0 = 0 and u ≤ 0, or m0 =M , let K ⊂ {1, . . . ,M − 1} be theset of hS indices. Then S+ = S + SK .

(2) If m0 = 0 and u > 0, take the sets of hS indices J ⊂ {0, . . . , t− 1}and K ⊂ {ut, . . . ,M − 1}. Then S+ = S + uSJ + nt (u−1)u

2 + SK .

(3) If 0 < m0 < M and m0 + t ≥M , take the sets of hS indicesI ⊂ {1, . . . ,m0 − 1} and K ⊂ {m0, . . . ,M − 1}. ThenS+ = S + SI + SK .

(4) If 0 < m0 < M and m0 + t < M , take the sets of hS indicesI ⊂ {1, . . . ,m0 − 1} and J ⊂ {m0,m0 + t− 1}. TakeK ⊂ {m0 + ut, . . . ,M − 1} whenever u > 0 and K = ∅ otherwise(thus SK = 0). Then S+ = S + SI + uSJ + nt (u−1)u

2 + SK .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 2 (t - q and gcd(t, q) = 1)Consider m0 ≡ q−1s(mod t) and u =

⌊M−m0−1

t

⌋. Then,

(1) If m0 = 0 and u ≤ 0, or m0 =M , let K ⊂ {1, . . . ,M − 1} be theset of hS indices. Then S+ = S + SK .

(2) If m0 = 0 and u > 0, take the sets of hS indices J ⊂ {0, . . . , t− 1}and K ⊂ {ut, . . . ,M − 1}. Then S+ = S + uSJ + nt (u−1)u

2 + SK .

(3) If 0 < m0 < M and m0 + t ≥M , take the sets of hS indicesI ⊂ {1, . . . ,m0 − 1} and K ⊂ {m0, . . . ,M − 1}. ThenS+ = S + SI + SK .

(4) If 0 < m0 < M and m0 + t < M , take the sets of hS indicesI ⊂ {1, . . . ,m0 − 1} and J ⊂ {m0,m0 + t− 1}. TakeK ⊂ {m0 + ut, . . . ,M − 1} whenever u > 0 and K = ∅ otherwise(thus SK = 0). Then S+ = S + SI + uSJ + nt (u−1)u

2 + SK .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 3 (t - q and gcd(t, q) = g > 1)

Set t = t/g. Take the set of hS indicesJ = {j1, . . . , jn} ⊂ {0, . . . , t− 1}. Set u =

⌊M−j1−1

t

⌋.

(1) If j1 ≥M , then S+ = S.(2) If 0 ≤ j1 < M , take the set of hS indices

K ⊂ {j1 + ut, . . . ,M − 1}. Then,

S+ = S + uSJ + nt(u− 1)u

2+ SK .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 3 (t - q and gcd(t, q) = g > 1)Set t = t/g. Take the set of hS indicesJ = {j1, . . . , jn} ⊂ {0, . . . , t− 1}. Set u =

⌊M−j1−1

t

⌋.

(1) If j1 ≥M , then S+ = S.(2) If 0 ≤ j1 < M , take the set of hS indices

K ⊂ {j1 + ut, . . . ,M − 1}. Then,

S+ = S + uSJ + nt(u− 1)u

2+ SK .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S+ sums

Theorem 3 (t - q and gcd(t, q) = g > 1)Set t = t/g. Take the set of hS indicesJ = {j1, . . . , jn} ⊂ {0, . . . , t− 1}. Set u =

⌊M−j1−1

t

⌋.

(1) If j1 ≥M , then S+ = S.(2) If 0 ≤ j1 < M , take the set of hS indices

K ⊂ {j1 + ut, . . . ,M − 1}. Then,

S+ = S + uSJ + nt(u− 1)u

2+ SK .

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Definition of S± sumsComputing S+ sums

Computing S± sums

Three similar results can be derived for computing S−(s, t, q,N).

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Denumerants from S± sums

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Generic comments

Sums S± enable numerical computation of denumerantsd(m, 〈a, b, c〉) with time cost O(m+ log c), in the worst case.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Generic comments

These sums also enable the obtention of closed expressions fordenumerants.

There are eight different ways for reducing a denumerant intoS± sums. This reduction depends on the following eightdifferent cases of δ = la−yb

c and θ = hb−wac of a given L-shape

H = L(l, h, w, y):(1) {δ = 0, w = 0},(2) {δ = 0, w 6= 0},(3) {θ = 0, y = 0},(4) {θ = 0, y 6= 0},(5) {δθ 6= 0, y = 0, w = 0},(6) {δθ 6= 0, y = 0, w 6= 0},(7) {δθ 6= 0, y 6= 0, w = 0},(8) and {δθ 6= 0, y 6= 0, w 6= 0}.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Generic comments

These sums also enable the obtention of closed expressions fordenumerants.

There are eight different ways for reducing a denumerant intoS± sums.

This reduction depends on the following eightdifferent cases of δ = la−yb

c and θ = hb−wac of a given L-shape

H = L(l, h, w, y):(1) {δ = 0, w = 0},(2) {δ = 0, w 6= 0},(3) {θ = 0, y = 0},(4) {θ = 0, y 6= 0},(5) {δθ 6= 0, y = 0, w = 0},(6) {δθ 6= 0, y = 0, w 6= 0},(7) {δθ 6= 0, y 6= 0, w = 0},(8) and {δθ 6= 0, y 6= 0, w 6= 0}.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Generic comments

These sums also enable the obtention of closed expressions fordenumerants.

There are eight different ways for reducing a denumerant intoS± sums. This reduction depends on the following eightdifferent cases of δ = la−yb

c and θ = hb−wac of a given L-shape

H = L(l, h, w, y):(1) {δ = 0, w = 0},(2) {δ = 0, w 6= 0},(3) {θ = 0, y = 0},(4) {θ = 0, y 6= 0},(5) {δθ 6= 0, y = 0, w = 0},(6) {δθ 6= 0, y = 0, w 6= 0},(7) {δθ 6= 0, y 6= 0, w = 0},(8) and {δθ 6= 0, y 6= 0, w 6= 0}.

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Consider S = 〈121, 1111, 2323〉 with related L-shapeH = L(101, 23, 0, 11). This is the case δ = w = 0.

Given m ∈ S, take his basic factorization (x0, y0, z0) withrespect to H. Then

d(m,S) = 1 +⌊z0θ

⌋+

b z0θ c∑k=0

(⌊y0 + k(h− y)

y

⌋+

⌊z0 − kθ

θ

⌋)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Consider S = 〈121, 1111, 2323〉 with related L-shapeH = L(101, 23, 0, 11). This is the case δ = w = 0.

Given m ∈ S, take his basic factorization (x0, y0, z0) withrespect to H.

Then

d(m,S) = 1 +⌊z0θ

⌋+

b z0θ c∑k=0

(⌊y0 + k(h− y)

y

⌋+

⌊z0 − kθ

θ

⌋)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Consider S = 〈121, 1111, 2323〉 with related L-shapeH = L(101, 23, 0, 11). This is the case δ = w = 0.

Given m ∈ S, take his basic factorization (x0, y0, z0) withrespect to H. Then

d(m,S) = 1 +⌊z0θ

⌋+

b z0θ c∑k=0

(⌊y0 + k(h− y)

y

⌋+

⌊z0 − kθ

θ

⌋)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Set y0 = y0y + y0 and h− y = ny + n with 0 ≤ y0, n < y. Setλ =

⌊z0θ

⌋.

Then

λ∑k=0

⌊y0 + k(h− y)

y

⌋= (1 + λ)(y0 +

1

2λn) +

λ∑k=0

⌊y0 + kn

y

⌋and

λ∑k=0

⌊z0 − kθ

θ

⌋=

λ∑k=0

(λ− k) = 1

2λ(1 + λ).

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Set y0 = y0y + y0 and h− y = ny + n with 0 ≤ y0, n < y. Setλ =

⌊z0θ

⌋. Then

λ∑k=0

⌊y0 + k(h− y)

y

⌋= (1 + λ)(y0 +

1

2λn) +

λ∑k=0

⌊y0 + kn

y

⌋and

λ∑k=0

⌊z0 − kθ

θ

⌋=

λ∑k=0

(λ− k) = 1

2λ(1 + λ).

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Putting y = 11, h = 23, n = 1, n = 1 and λ =⌊z011

⌋d(m,S) = (1 +

⌊ z011

⌋)(1 + y0 +

⌊ z011

⌋) +

b z011c∑k=0

⌊y0 + k

11

⌋.

As t | q (t = 1 and q = 11), this discrete sum can be computedby Theorem 1

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

Putting y = 11, h = 23, n = 1, n = 1 and λ =⌊z011

⌋d(m,S) = (1 +

⌊ z011

⌋)(1 + y0 +

⌊ z011

⌋) +

b z011c∑k=0

⌊y0 + k

11

⌋.

As t | q (t = 1 and q = 11), this discrete sum can be computedby Theorem 1

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

d(m,S) =(1 +⌊ z011

⌋)(1 +

⌊ z011

⌋+ y0 +Mz0)

−Mz0 [11

2(Mz0 + 1)− y0],

where Mz0 =

⌊y0+b z011c

11

⌋and the basic factorization (x0, y0, z0)

of m with respect to H are obtained at time cost O(log c), in theworst case.

Corollary Assume m ∼ (x0, y0, z0) and m′ ∼ (x′0, y′0, z′0) are the

basic factorizations with respect to H. Theny0 = y′0, bz0/11c = bz′0/11c ⇒ d(m,S) = d(m′, S)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

Reduction example

d(m,S) =(1 +⌊ z011

⌋)(1 +

⌊ z011

⌋+ y0 +Mz0)

−Mz0 [11

2(Mz0 + 1)− y0],

where Mz0 =

⌊y0+b z011c

11

⌋and the basic factorization (x0, y0, z0)

of m with respect to H are obtained at time cost O(log c), in theworst case.

Corollary Assume m ∼ (x0, y0, z0) and m′ ∼ (x′0, y′0, z′0) are the

basic factorizations with respect to H. Theny0 = y′0, bz0/11c = bz′0/11c ⇒ d(m,S) = d(m′, S)

Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona

IntroductionDistribution of 3–factorizations

S+ and S− sumsDenumerants from S± sums

Generic commentsReduction example

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Denumerant of numer. 3–semigroups IMNS–2014 Il Palazzone, Cortona