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A370235 Table read by rows. Number of set partitions of [n] with respect to genus g.

%I A370235 #51 Feb 23 2024 08:03:43
%S A370235 1,1,2,5,14,1,42,10,132,70,1,429,420,28,1430,2310,399,1,4862,12012,
%T A370235 4179,94,16796,60060,36498,2620,1,58786,291720,282282,45430,352,
%U A370235 208012,1385670,1999998,600655,19261,1,742900,6466460,13258674,6633484,541541,1378
%N A370235 Table read by rows. Number of set partitions of [n] with respect to genus g.
%C A370235 The table shows the number of partitions of [n] = {1, 2, 3, ..., n} with genus g.
%C A370235 The set of noncrossing partitions is exactly the set of genus zero partitions. The numbers corresponding to this case are the Catalan numbers.
%C A370235 This is essentially table 2.1 in Martha Yip's thesis (p. 12).
%C A370235 From _Robert Coquereaux_, Feb 16 2024: (Start)
%C A370235 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.
%C A370235 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.
%C A370235 (End)
%H A370235 Robert Coquereaux, <a href="/A370235/b370235.txt">Table of n, a(n) for n = 0..57</a> (rows 0..15)
%H A370235 Martha Yip, <a href="https://uwspace.uwaterloo.ca/handle/10012/2933">Genus one partitions</a>, Master Thesis, University of Waterloo, 2006. [Typos in Table 2.1 in positions T(8, 0) and T(10, 0)].
%H A370235 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 4, 5, 22. Also in <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Coquereaux/coque5.html">JIS</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 8, 9, 10, 32.
%e A370235 [n\g]     0        1        2      3      4     5
%e A370235 -------------------------------------------------
%e A370235 [ 0]      1;
%e A370235 [ 1]      1;
%e A370235 [ 2]      2;
%e A370235 [ 3]      5;
%e A370235 [ 4]     14,       1;
%e A370235 [ 5]     42,      10;
%e A370235 [ 6]    132,      70,        1;
%e A370235 [ 7]    429,     420,       28;
%e A370235 [ 8]   1430,    2310,      399,       1;
%e A370235 [ 9]   4862,   12012,     4179,      94;
%e A370235 [10]  16796,   60060,    36498,    2620,      1;
%e A370235 [11]  58786,  291720,   282282,   45430,    352;
%e A370235 [12] 208012, 1385670,  1999998,  600655,  19261,    1;
%e A370235 [13] 742900, 6466460, 13258674, 6633484, 541541, 1378;
%Y A370235 Columns: A000108 (g=0), A002802 (g=1), A297179 (g=2), A370237 (g=3).
%Y A370235 Cf. A000110 (row sums), A177267 (permutations by genus).
%Y A370235 Cf. A001263, A370236, A297178.
%Y A370235 Cf. A370420 (S2(n,k,g)).
%K A370235 nonn,tabf,hard
%O A370235 0,3
%A A370235 _Peter Luschny_, Feb 15 2024