cp's OEIS Frontend

A370420 Number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). Flattened 3-dimensional array read by n, then by g:0..floor(n-1)/2, then by k:1..n.

%I A370420 #37 Feb 21 2024 03:46:03
%S A370420 1,1,1,1,3,1,1,6,6,1,0,1,0,0,1,10,20,10,1,0,5,5,0,0,1,15,50,50,15,1,0,
%T A370420 15,40,15,0,0,0,1,0,0,0,0,1,21,105,175,105,21,1,0,35,175,175,35,0,0,0,
%U A370420 7,21,0,0,0,0,1,28,196,490,490,196,28,1,0,70,560,1050,560,70,0,0,0,28,210,161,0,0,0,0,0,1,0,0,0,0,0,0
%N A370420 Number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). Flattened 3-dimensional array read by n, then by g:0..floor(n-1)/2, then by k:1..n.
%C A370420 Genus-dependent Stirling numbers of the second kind S2(n,k,g), 1 <= n, 1 <= k <= n, 0 <= g <= floor((n-1)/2). This is an infinite three-dimensional array. Its first 15 rows (n:1..15) are given by the table (see Links) taken from the article by Robert Coquereaux and Jean-Bernard Zuber (where a transpose of this table is given), see p. 32. These 15 rows determine 589 entries of the sequence (Data).
%C A370420 Example: the numbers S2(5,k,0), k=1..5, are {1,10,20,10,1} and appear on line 5, column 1; the numbers S2(5,k,1), k=1..5, are {0,5,5,0,0} and appear on line 5, column 2. Values of S2(n,k,g) for g > floor((n-1)/2) are equal to 0 and are not displayed.
%C A370420 Summing S2(n,k,g) over k gives genus-dependent Bell numbers B(n,g), A370235. Summing S2(n,k,g) over g gives S2(n,k), the Stirling numbers of the second kind A008277. Summing S2(n,k,g) over k and g gives the Bell numbers B(n), A000110. Example: S2(5,k,0) = 1, 10, 20, 10, 1 and S2(5,k,1) = 0, 5, 5, 0, 0 for k = 1..5; therefore S2(5,k) = 1, 15, 25, 10, 1, B(5,0) = 42, B(5,1) = 10, and B(5) = 52.
%H A370420 Robert Coquereaux, <a href="/A370420/b370420.txt">Rows n = 1..15 of array, flattened, terms 1..589</a>
%H A370420 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.
%H A370420 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Coquereaux/coque5.html">Counting partitions by genus: a compendium of results</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 8, 9, 10, 32.
%H A370420 Robert Coquereaux, <a href="/A370420/a370420.txt">Table of S2(n,k,g) for n = 1..15</a>
%H A370420 Robert Coquereaux, <a href="/A370420/a370420_1.txt">Mathematica program for the numbers S(n,k,g) and B(n,g)</a>
%H A370420 Robert Cori and Gabor Hetyei, <a href="https://doi.org/10.37236/7632">Counting partitions of a fixed genus</a>, Electron. J. Combin. 25 (4) (2018), #P4.26.
%H A370420 Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2303.05875">Counting partitions by genus. I. Genus 0 to 2</a>, arXiv:2303.05875 [math.CO], 2023.
%H A370420 Jean-Bernard Zuber, <a href="https://doi.org/10.54550/ECA2024V4S2R13">Counting partitions by genus. I. Genus 0 to 2</a>, Enumer. Comb. Appl. 4 (2) (2024) #S2R13.
%F A370420 No general formula is currently known. In the particular cases g=0, 1, 2, a formula is known: see Crossrefs.
%e A370420 For n:1..7, g:1..floor(n-1)/2, k:1..n. The 3-dimensional array begins:
%e A370420   {1};
%e A370420   {1,1};
%e A370420   {1,3,1};
%e A370420   {1,6,6,1},               {0,1,0,0};
%e A370420   {1,10,20,10,1},          {0,5,5,0,0};
%e A370420   {1,15,50,50,15,1},       {0,15,40,15,0,0},      {0,1,0,0,0,0};
%e A370420   {1,21,105,175,105,21,1}, {0,35,175,175,35,0,0}, {0,7,21,0,0,0,0};
%t A370420 See Links
%Y A370420 Cf. A001263 (g=0), A370236 (g=1), A297178 (g=2).
%Y A370420 Cf. A370235 (sum over k).
%Y A370420 Cf. A000110, A008277.
%K A370420 nonn,tabf
%O A370420 1,5
%A A370420 _Robert Coquereaux_, Feb 18 2024