cp's OEIS Frontend

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).

%I A297178 #39 Mar 05 2024 14:40:28
%S A297178 1,7,21,28,210,161,84,1134,2184,777,210,4410,15330,13713,2835,462,
%T A297178 13860,75075,121275,63063,8547,924,37422,289905,729960,685608,233772,
%U A297178 22407,1716,90090,942942,3396393,4972968,3063060,738738,52767,3003,198198,2690688,13096083,27432405,26342316,11477466,2063061,114114
%N 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).
%H A297178 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. 7.
%H A297178 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. 12.
%H A297178 Robert Cori and G. Hetyei, <a href="https://arxiv.org/abs/1710.09992">Counting partitions of a fixed genus</a>, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
%H A297178 Martha Yip, <a href="https://uwspace.uwaterloo.ca/handle/10012/2933">Genus one partitions</a>, Master Thesis, University of Waterloo, 2006.
%F A297178 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
%F A297178 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
%e A297178 Triangle begins (see Table 3.2 in Yip's thesis):
%e A297178     1;
%e A297178     7,    21;
%e A297178    28,   210,    161;
%e A297178    84,  1134,   2184,    777;
%e A297178   210,  4410,  15330,  13713,   2835;
%e A297178   462, 13860,  75075, 121275,  63063,   8547;
%e A297178   924, 37422, 289905, 729960, 685608, 233772, 22407;
%e A297178   ...
%t A297178 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 *)
%Y A297178 Row sums are A297179.
%Y A297178 First column is A000579.
%K A297178 nonn,tabl
%O A297178 6,2
%A A297178 _N. J. A. Sloane_, Dec 26 2017