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).

1, 5, 5, 15, 40, 15, 35, 175, 175, 35, 70, 560, 1050, 560, 70, 126, 1470, 4410, 4410, 1470, 126, 210, 3360, 14700, 23520, 14700, 3360, 210, 330, 6930, 41580, 97020, 97020, 41580, 6930, 330, 495, 13200, 103950, 332640, 485100, 332640, 103950, 13200, 495

Offset: 4

Author

Robert Coquereaux, Feb 12 2024

Keywords

• nonn
• tabl

Comments

The formula given below was conjectured by Martha Yip and proved by Robert Cori and Gábor Hetyei.

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.

Then T(n, k) = S(n, k, 1). See Robert Coquereaux and Jean-Bernard Zuber.

Examples

			Triangle begins (see Table 3.1 in Yip's thesis):
    1;
    5,    5;
   15,   40,   15;
   35,  175,  175,   35;
   70,  560, 1050,  560,   70;
  126, 1470, 4410, 4410, 1470, 126;
		

Links

• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 9. See also arXiv:2305.01100, 2023.
• Robert Cori and Gábor Hetyei, Counting genus one partitions and permutations, arXiv:1306.4628 [math.CO], 2013.
• Robert Cori and Gábor Hetyei, Counting genus one partitions and permutations, Sémin. Lothar. Comb. 70, B70e, 30 p. (2014).
• Martha Yip, Genus one partitions, Master Thesis, University of Waterloo, 2006.

Crossrefs

Row sums are A002802.

Cf. A000332, A297178 (genus 2).

Programs

Mathematica

T[n_,k_] := (1/6) Binomial[n, 2] Binomial[n-2, k] Binomial[n-2, k-2];
Table[T[n,k],{n,4,12},{k,2,n-2}]//Flatten (* Stefano Spezia, Feb 14 2024 *)

Formula

T(n, k) = (1/6)*binomial(n, 2)*binomial(n-2, k)*binomial(n-2, k-2).