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A370236 Triangle read by rows: T(n, k) is the number of partitions of genus 1 and k parts of the n-set (n >= 4, 2 <= k <= n-2).

%I A370236 #25 Feb 16 2024 04:28:03
%S A370236 1,5,5,15,40,15,35,175,175,35,70,560,1050,560,70,126,1470,4410,4410,
%T A370236 1470,126,210,3360,14700,23520,14700,3360,210,330,6930,41580,97020,
%U A370236 97020,41580,6930,330,495,13200,103950,332640,485100,332640,103950,13200,495
%N A370236 Triangle read by rows: T(n, k) is the number of partitions of genus 1 and k parts of the n-set (n >= 4, 2 <= k <= n-2).
%C A370236 The formula given below was conjectured by Martha Yip and proved by Robert Cori and Gábor Hetyei.
%C A370236 More generally one may consider genus-dependent Stirling numbers S(n, k, g) that count the partitions of genus g and k parts of the n-set.
%C A370236 Then T(n, k) = S(n, k, 1). See Robert Coquereaux and Jean-Bernard Zuber.
%H A370236 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. 9. See also arXiv:2305.01100, 2023.
%H A370236 Robert Cori and Gábor Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv:1306.4628 [math.CO], 2013.
%H A370236 Robert Cori and Gábor Hetyei, <a href="https://www.elibm.org/article/10009581">Counting genus one partitions and permutations</a>, Sémin. Lothar. Comb. 70, B70e, 30 p. (2014).
%H A370236 Martha Yip, <a href="https://uwspace.uwaterloo.ca/handle/10012/2933">Genus one partitions</a>, Master Thesis, University of Waterloo, 2006.
%F A370236 T(n, k) = (1/6)*binomial(n, 2)*binomial(n-2, k)*binomial(n-2, k-2).
%e A370236 Triangle begins (see Table 3.1 in Yip's thesis):
%e A370236     1;
%e A370236     5,    5;
%e A370236    15,   40,   15;
%e A370236    35,  175,  175,   35;
%e A370236    70,  560, 1050,  560,   70;
%e A370236   126, 1470, 4410, 4410, 1470, 126;
%t A370236 T[n_,k_] := (1/6) Binomial[n, 2] Binomial[n-2, k] Binomial[n-2, k-2];
%t A370236 Table[T[n,k],{n,4,12},{k,2,n-2}]//Flatten (* _Stefano Spezia_, Feb 14 2024 *)
%Y A370236 Row sums are A002802.
%Y A370236 Cf. A000332, A297178 (genus 2).
%K A370236 nonn,tabl
%O A370236 4,2
%A A370236 _Robert Coquereaux_, Feb 12 2024