A297178 Triangle read by rows: T(n,k) = number of partitions of genus 2 of n elements with k parts (n >= 6, 2 <= k <= n-4).
1, 7, 21, 28, 210, 161, 84, 1134, 2184, 777, 210, 4410, 15330, 13713, 2835, 462, 13860, 75075, 121275, 63063, 8547, 924, 37422, 289905, 729960, 685608, 233772, 22407, 1716, 90090, 942942, 3396393, 4972968, 3063060, 738738, 52767, 3003, 198198, 2690688, 13096083, 27432405, 26342316, 11477466, 2063061, 114114
Offset: 6
Examples
Triangle begins (see Table 3.2 in Yip's thesis):
1;
7, 21;
28, 210, 161;
84, 1134, 2184, 777;
210, 4410, 15330, 13713, 2835;
462, 13860, 75075, 121275, 63063, 8547;
924, 37422, 289905, 729960, 685608, 233772, 22407;
...
Links
- Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 7.
- 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. 12.
- Robert Cori and G. Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
- Martha Yip, Genus one partitions, Master Thesis, University of Waterloo, 2006.
Programs
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Mathematica
T[n_,k_]:=((-6*(-2 + n)*(-1 + n) - k^2*(-13 + 5*n) + k*(-8 + n*(-9 + 5*n)))*(-4 + n)!*n!)/(1440*(-2 + k)!*k!*(-4 - k + n)!*(-k + n)!) (* Robert Coquereaux, Mar 05 2024 *)
Formula
T(n,k) = 8*gam(n-10,k-6) -4*gam(n-10,k-5) -15*gam(n-10,k-4) +10*gam(n-10,k-3) +gam(n-10,k-2) -4*gam(n-9,k-5) +39*gam(n-9,k-4) -10*gam(n-9,k-3) -4*gam(n-9,k-2) -15*gam(n-8,k-4) -10*gam(n-8,k-3) +6*gam(n-8,k-2) -4*gam(n-7,k-2) +10*gam(n-7,k-3) +gam(n-6,k-2) with gam(n,k) = (binomial(n+10,5) * binomial(n+5,k) * binomial(n+5,n-k)) / binomial(10,5) [Cori & Hetyei]. - Robert Coquereaux, Feb 12 2024
T(n,k) = ((-6*(-2 + n)*(-1 + n) - k^2*(-13 + 5*n) + k*(-8 + n*(-9 + 5*n)))*(-4 + n)!*n!) / (1440*(-2 + k)!*k!*(-4 - k + n)!*(-k + n)!). - Robert Coquereaux, Mar 05 2024