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hivert / NumericMonoid

Computing the number of Numerical Monoid of a Given Genus

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Computing the number of Numerical Monoid of a Given Genus

This is a very optimized implementation of algorithm described in

Jean Fromentin and Florent Hivert. 2016. Exploring the tree of numerical semi- groups. Math. Comput. 85, 301 (2016), 2553–2568. DOI:https://doi.org/10.1090/mcom/3075

The more up to date code is in directory src/Cilk++/ together with a Sagemath binding.

Description of the problem

A numerical semigroup is a subset of the set of natural number which

• contains 0
• is stable under addition
• has a finite complement

The elements of the complement are called gaps. The number of gaps is called the genus.

The goal is to compute the number n(g) of semigroups of a given genus.

A few conjectures:

• Bras-amoros 2008 : n(g) >= n(g-1) + n(g-2). Still widely open.
• Zhai 2013 n(g) >= n(g-1) asymptotically true, but open for small g.

See http://images.math.cnrs.fr/Semigroupes-numeriques-et-nombre-d-or-II.html (in French) for more explanation.

We also validated Wilf conjecture upto n=60 and invalidated some stronger statements (See Near-misses in Wilf's conjecture, Shalom Eliahou, Jean Fromentin https://arxiv.org/abs/1710.03623v1).

Results

Below is the table of the results (A more computer friendly syntax is at the end of https://github.com/hivert/NumericMonoid/raw/master/src/Sizes

g number of semigroups
0 1
1 1
2 2
3 4
4 7
5 12
6 23
7 39
8 67
9 118
10 204
11 343
12 592
13 1001
14 1693
15 2857
16 4806
17 8045
18 13467
19 22464
20 37396
21 62194
22 103246
23 170963
24 282828
25 467224
26 770832
27 1270267
28 2091030
29 3437839
30 5646773
31 9266788
32 15195070
33 24896206
34 40761087
35 66687201
36 109032500
37 178158289
38 290939807
39 474851445
40 774614284
41 1262992840
42 2058356522
43 3353191846
44 5460401576
45 8888486816
46 14463633648
47 23527845502
48 38260496374
49 62200036752
50 101090300128
51 164253200784
52 266815155103
53 433317458741
54 703569992121
55 1142140736859
56 1853737832107
57 3008140981820
58 4880606790010
59 7917344087695
60 12841603251351
61 20825558002053
62 33768763536686
63 54749244915730
64 88754191073328
65 143863484925550
66 233166577125714
67 377866907506273
68 612309308257800
69 992121118414851
70 1607394814170158

Computing the number of Numerical Monoid of a Given Genus

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