OFFSET
0,3
COMMENTS
The table shows the number of partitions of [n] = {1, 2, 3, ..., n} with genus g.
The set of noncrossing partitions is exactly the set of genus zero partitions. The numbers corresponding to this case are the Catalan numbers.
This is essentially table 2.1 in Martha Yip's thesis (p. 12).
From Robert Coquereaux, Feb 16 2024: (Start)
The two-dimensional array is called triangle of genus-dependent Bell numbers B(n, g); if n >= 1, n even, nonzero values are obtained for 0 <= g <= floor((n-1)/2); if n >= 1, odd, nonzero values are obtained for 0 <= g < (n-1)/2.
The two-dimensional array B(n, g) can be obtained from a three-dimensional array S2(n, k, g), by summation over the number k of blocks. The numbers S2(n, k, g) are genus-dependent Stirling numbers of the second kind. They give the number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). The numbers S2(n, k, g) are discussed in A370420.
(End)
LINKS
Robert Coquereaux, Table of n, a(n) for n = 0..57 (rows 0..15)
Martha Yip, Genus one partitions, Master Thesis, University of Waterloo, 2006. [Typos in Table 2.1 in positions T(8, 0) and T(10, 0)].
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 4, 5, 22. Also in JIS, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 8, 9, 10, 32.
EXAMPLE
[n\g] 0 1 2 3 4 5
-------------------------------------------------
[ 0] 1;
[ 1] 1;
[ 2] 2;
[ 3] 5;
[ 4] 14, 1;
[ 5] 42, 10;
[ 6] 132, 70, 1;
[ 7] 429, 420, 28;
[ 8] 1430, 2310, 399, 1;
[ 9] 4862, 12012, 4179, 94;
[10] 16796, 60060, 36498, 2620, 1;
[11] 58786, 291720, 282282, 45430, 352;
[12] 208012, 1385670, 1999998, 600655, 19261, 1;
[13] 742900, 6466460, 13258674, 6633484, 541541, 1378;
CROSSREFS
Cf. A370420 (S2(n,k,g)).
KEYWORD
nonn,tabf,hard
AUTHOR
Peter Luschny, Feb 15 2024
STATUS
approved