# [seqfan] seqs from "Computing the number of numerical semigroups using generating functions"

Jonathan Post jvospost3 at gmail.com
Sat Jan 9 00:45:39 CET 2010

```There are several seqs to extract from table 1 on p.5 of
http://arxiv.org/abs/0901.1228

Computing the number of numerical semigroups using generating functions
Authors: Victor Blanco, Pedro A. Garcia-Sanchez, Justo Puerto
(Submitted on 9 Jan 2009 (v1), last revised 23 Dec 2009 (this version, v3))

Abstract: This paper presents a new methodology to count the
number of numerical semigroups of given genus or Frobenius number. We
apply generating function tools to the bounded polyhedron that
classifies the semigroups with given genus (or Frobenius number) and
multiplicity. First, we give theoretical results about the
polynomial-time complexity of counting the number of these semigroups.
We also illustrate the methodology analyzing the cases of multiplicity
3 and 4 where some formulas for the number of numerical semigroups for
any genus and Frobenius number are obtained.

g\m 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...........................n_g
2 1 1 .................................................................................2
3 1 2 1 ..............................................................................4
4 1 2 3 1 ...........................................................................7
5 1 2 4 4 1 .......................................................................12
6 1 3 6 7 5 1 ....................................................................33
7 1 3 7 10 11 6 1
..............................................................39
8 1 3 9 13 17 16 7 1
..........................................................67
9 1 4 11 16 27 28 22 8 1 ..................................................118
10 1 4 13 22 37 44 44 29 9 1 ............................................204
11 1 4 15 24 49 64 72 66 37 10 1 ......................................343
12 1 5 18 32 66 85 116 116 95 46 11 1 ..............................592
13 1 5 20 35 85 112 172 188 182 132 56 12 1 ...................1001
14 1 5 23 43 106 148 239 288 304 277 178 67 13 1 ...........1693
15 1 6 26 51 133 191 325 409 492 486 409 234 79 14 1 .... 2857
Table 1. Number of numerical semigroups with given genus g and multiplicity m.

the m=2 column == the g =m-1 diagonal == A000012 == all ones
the g = m diagonal == A000027 == n
the g = m+1 diagonal == A000124 Central Polygonal Numbers
the g = m+2 diagonal is new: 1, 2, 6, 10, 17, 28, 44, 66, 95, 132, 178, 234
the m= 3 column is new
the m=4 column is new 1, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, ...
and so for other rows and columns
the row sums n_g (2, 4, 7, 12, 33, 39, 67, 118, 204, 343, 592, 1001,
1693, 2857, ...) is new

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